Answer a question that qualifies as a tumbleweed and get that answer accepted
The hat description pretty much explains everything. Like several other hat names this year, it's a play on words; the term "weed eater" itself is a genericized trademark that refers to a type of lawn care tool.
I'm sad to say I've never heard of The Dark Tower or The Gunslinger, let alone the specific event from the story that you're thinking of.
Vote or post on Dec. 14 (the first day of Winter Bash, and the last day of Hanukkah)
This one was pretty straightforward. Sufganiyot are donuts traditionally eaten during Hanukkah. Thanks to Double AA and others at Mi Yodeya (AKA Jewish Life and Learning SE) for confirming that these are a good symbol of the holiday.
A New Hope
Vote or post on Dec. 18 (the release date of Star Wars: The Force Awakens in the US)
I have to say, I expected at least a few complaints about this hat name from people who didn't get the joke, but I didn't see any, so either it was both obvious and hilarious or people got it but didn't think it was funny. It's the first one, right? Right?
For those who aren't aware (don't worry, I won't judge... much): A New Hope is the title of the original Star Wars movie, released in May 1977. The three prequel movies, released two decades later, were... not exactly loved... by fans, and the new film released last month was seen as something of a new hope for the franchise.
The Airing of Grievances
Vote or post on meta on Dec. 23 (Festivus)
Who loves the '90s? One of the most famous and popular  episodes of Seinfeld, "The Strike", introduced millions of people to the holiday of Festivus, celebrated every Dec. 23 with an aluminum pole, a display of feats of strength, and of course, an airing of grievances. And what better place to air your grievances than meta?
Vote or post on Dec. 25 (Christmas)
Another straightforward one. There's a famous Christmas carol titled O Tannenbaum, also known as O Christmas Tree ("tannenbaum" being the German word for a fir tree, and often used to refer to Christmas trees).
Auld Lang Syne
Post in chat within 12 hours of the moment when 2015 becomes 2016 (UTC), and get a star
Continuing the musical theme, Auld Lang Syne is a song traditionally sung to celebrate the arrival of a new year. According to Wikipedia,
The song's Scots title may be translated into standard English as "old long since", or more idiomatically, "long long ago", "days gone by" or "old times". Consequently, "For auld lang syne", as it appears in the first line of the chorus, might be loosely translated as "for (the sake of) old times".
Visit the site on ten consecutive days during Winter Bash
Visiting a site for 100 consecutive days earns a user the Fanatic gold badge. Mostly, I couldn't resist the pun here. This is also partially to throw a bone to those people who have complained that it's too hard to achieve the badge, or even its 30-day counterpart, Enthusiast. (But if you think 100 is hard... have you seen these guys?)
The hat design, incidentally, was originally requested here on meta, albeit with a different trigger. The requested trigger ended up being used for another hat this year, Sun Wukong (scroll down).
Every! Body! Gets! A Hat!
Participate in Winter Bash on a site where at least 20 hats have been unlocked
This one makes sense if and only if you get the reference. Back in 2004, Oprah Winfrey taped an episode of her eponymous show in which every member of the audience received a car as a free gift. Check out the video, then come back and read the name of the hat in her voice.
Earn 11 other hats
Here's a rare instance of the community guessing the origin of a hat design incorrectly, although based on the screenshot provided, I can totally see why.
This year's hat isn't an American football helmet. We did a football helmet last year, as the Handegg hat:
(Bonus naming info: What's a "Handegg"?, at English Language & Usage SE; and this answer in particular)
This year's hat is, as mmyers said, a cricket batting helmet. The name of the hat is a play on "specialist batsman," the term for a cricketer who, well, specializes in batting (as opposed to bowling). The trigger required 11 other hats because there are 11 players on a cricket team. And the unicorn in the helmet logo (both this year and last) isn't just some rando off the street (or, uh, sky pasture?), it's Sparkles, our unofficial company pet/mascot.
Receive no downvotes and at least five upvotes in a day
Carl Fredricksen is the name of the main character from one of the biggest movies of 2009. The name of the film? Up.
Do You Even Lift?
Get at least five answers to scores of +5 each
The phrase "Do you even lift?" comes to us from an Internet meme. (If you like memes, scroll down to the secret hat section and check out It's Over 9000!) The trigger was chosen to evoke five sets of five reps. And what better "hat" to represent this than a sweatband?
Side note: thousands of these were earned, so maybe I just missed it, but I'm disappointed that I didn't see anyone abusing the opportunity to look like a French mayor.
Living in the Future
Post a self-answer that scores at least 5 to a question asked before Winter Bash started
As a certain segment of the population will undoubtedly remember, Back to the Future Day occurred in 2015. What better way to commemorate this than two months late, with Marty McFly's outfit from the second movie?
Do it Yourself
Ask and self-answer a question; each post must reach score of at least 3
Here's another mostly self-explanatory one. Self-answering is "doing it yourself." "Do it yourself" is also a common phrase, at least in the US, referring to a wide variety of projects people can do on their own—often relating to home improvement—instead of buying a mass-produced product or hiring a professional.
Answer within 30 minutes of question being asked, scoring at least 3 and getting accepted
Users of a certain age, at least in North America, will immediately recognize that this is a reference to the children's TV show Mister Rogers' Neighborhood. Specifically, the character Mr. McFeely, the delivery man, who worked for the Speedy Delivery company and said "Speedy Delivery!" whenever he arrived to make a delivery.
For a question with an existing accepted answer, post a competing answer that reaches a score of at least +3
This hat was named for Copernicus in recognition of his theory of heliocentrism, which challenged an existing solution to the question of how astronomical objects moved (geocentrism) that people had already accepted as true, and was eventually rewarded by widespread support.
It was also a bit of a test for intrepid Community Manager bluefeet, as part of ongoing research into the worst prominently displayed answers.
Successfully close a question
Melpomene is the Greek Muse of Tragedy. Mostly I just thought it would be funny to use the drama and comedy masks as hats. Closing is a sad face occasion, hence Melpomene.
Successfully reopen a question
Thalia is the Greek Muse of Comedy. Mostly I just thought it would be funny to use the comedy and drama masks as hats. Reopening is a happy face occasion, hence Thalia.
Cerro de Potosi
Earn a silver badge
Like the other "badge hat" (El Dorado, below), this is pretty straightforward. Cerro de Potosi (AKA Cerro Rico) is the most famous silver mine in the world. At one point, it was rumored that the entire mountain was basically made of silver.
Earn a gold badge
Like the other "badge hat" (Cerro de Potosi, above), this is pretty straightforward. El Dorado is a mythical city made of gold, long searched for in the Americas by Spanish (and other) explorers.
Post on a new site for the first time and reach a score of 3
Another fairly self-explanatory hat. This is for "exploring" new places in the SE network. It's one of my favorite hat ideas of all time (Hello, World in 2013 was very similar). The hat itself is a pith helmet, often associated in the modern day with European explorers journeying to tropical locales.
Edit and upvote a new user's first post
With all those Explorers running around, we needed welcoming committees for them! The name and art from this hat actually came from the job title and uniform of people who stand just inside the doors of Wal-Marts to welcome customers.
Earn the Nice Question badge for a post made from the mobile app
Some of us on the team have a use case story in our heads for the mobile app. You're in the kitchen making dinner, or out back pruning a rose bush, and you run into a problem. Maybe you need to keep an eye on a sauce that's reducing and can't leave for long; maybe you just don't feel like dropping everything you're doing in the garden and breaking your mental groove to go find a computer.
Either way, you whip out your phone, bang out a quick question in the app, maybe snap a quick photo, and throw to Seasoned Advice or Gardening and Landscaping SE for help from "in the field."
Certainly that's not the only use case for the app; it's just one possible story. But it's one I thought might be nice to reward with a hat.
Vote ten times from the mobile app
This one is not only straightforward, but also boring. Mobile devices are wireless, so the hat is a wireless headset. What? They can't all come with great stories.
Instead, I leave you with a pro tip: we got reports this year that emulators work just fine for mobile-related hats, for those who are unwilling or unable to install the app on real mobile devices.
Edit a closed question (that you did not participate in closing) that gets reopened after your edit
Probably the most straightforward hat of the year. Actual comment from our brainstorming doc:
How have we not done a Batman hat yet?
Edit five old questions
Last year, we had a hat named Time Lord with the exact same trigger that looked like a TARDIS. It was widely liked, and given the nature of both the trigger and Dr. Who, I thought running it again with just a slight change seemed appropriate.
Hit the rep cap three times
Beyond the obvious element of "how could we not have a hat with 'hat' in the name," "hat trick" is a generic term in sports for achieving something three times, usually having to do with scoring points. I'm most familiar with it in the context of ice hockey, so the Winter Bash hat was an ice hockey helmet and jersey.
Use the site search on three consecutive days
This was a rare case of a hat being designed to fit the art rather than the other way around. Someone on the team declared that we needed to have a hat that looked like Taco wearing nerd glasses, and it quickly became accepted as fact.
No, not tacos, the food. Taco, the Siberian husky who lives with our CEO, Joel Spolsky. You can see a baby picture of him (Taco, not Joel) here, and a somewhat more recent one here, if you're into that sort of thing. A few people thought this hat was based on the Trello logo. Close, but not quite; their mascot and the hat just look similar because they were based on the same actual dog.
Vote Early, Vote Often
Cast 250 total votes on any seven consecutive days during Winter Bash
Historically, this phrase first became prominent in the context of political corruption in Chicago in the early 20th century. It's had a long history at Stack Exchange in less corrupt, more literal usage, though, showing up in the blog and on Math SE's meta from just a five-second Googling.
Guess the trigger for a secret hat other than Archimedes itself
This hat was originally rolled out in 2013, with the name "Eureka" and a light bulb as the design. Last year, the name remained the same and the art was tweaked only very slightly. I decided that this was not a tradition worth keeping, but I didn't want to throw away our history entirely, so this year's iteration was named after the man most famous for shouting "Eureka!" in the first place. The hat is an upside-down mini-bathtub because he did it while in the tub, or so the story goes.
Receive three anonymous positive feedback "pseudo-upvotes" on a question
This might have been the most misleading name of the year; it was certainly the most baffling trigger, as it was the only one that nobody earned Archimedes for. Almost all the way through the three weeks of Winter Bash, most people—at least, the ones who cared about such things—were divided into two camps: those who thought the name referred to the Final Fantasy job class and those who thought it referred to the Game of Thrones character. (A couple people on the fringes mentioned a Dark Souls NPC, too.)
In the wrap-up blog post, I said they were all wrong, but that's not 100% accurate. I wouldn't have picked this name for the hat if Final Fantasy and Game of Thrones hadn't made the phrase "Onion Knight" popular in the first place. But that's as far as the connection goes. The real hint in the trigger was the first word, "onion." Onions were meant to make people think of Tor, which in turn should have prompted thoughts about anonymity. (To be clear, actually using Tor itself was not required for the hat. Tracking Tor usage would actually be sort of creepy.)
Post or vote on Dec. 22 (the solstice)
This ended up being another misleading hat name, but quite by accident. Due to the way Winter Bash handles time, most people started seeing this hat being worn by others when the date was still 12/21 (or 21/12) in their local time zone. As a result, there were a lot of incorrect theories posted about the hat having to do with the palindromic date.
Actually, the thing that was "flip-flopping" was the length of days (or, rather, the sunlit parts of days), since the hat was awarded on the solstice. To a lesser degree, the name also referred to the hat design—the photo of the dress that the entire Internet flipped its lid (sorry) about earlier in the year—because some people were able to "flip-flop" between seeing the different color schemes.
Reply to Jon Ericson
This is actually a pretty literal hat name, but one that only serious Winter Bash fans would understand. Two years ago, Abby AKA "hairboat" was in charge of Winter Bash. Last year, Jon Ericson had the job, and he "decided to
prank her honor her with a hat awarded to those responding to one of her posts or comments". This year, Abby decided to get revenge by having us secretly run the hairboat hat again, only with Jon's user ID replacing hers in the trigger. We even used the same art as last year; the only change was switching the color of the sail to black, a nod to the Dread Pirate Roberts's ship Revenge, from The Princess Bride.
Answer a question scoring -3 or worse; the answer must score at least 5, and the question must improve to 3 or better; your votes do not count
After the success of the Red Baron hat last year, we wanted to run it again. But we wanted to correct one minor loophole from last year, too: preventing an answerer's own votes from counting towards the question improvement requirement.
In doing so, we made the hat harder to earn than Red Baron, and so we needed a name that was similar but better. Unfortunately, there aren't many pilots who have topped Manfred von Richthofen, by any measure. In World War II, Erich Hartmann became the top flying ace of all time, but he flew for Germany, and I didn't think that would make for a very good Winter Bash hat for obvious reasons. Eventually, I gave up on finding a strictly superior hat name and went for a lateral move, to something that was still in the same realm (military aviation) but not directly comparable.
Delete ten of your own comments after the post they're under was edited by its author
Another straightforward one, which is actually somewhat rare for a secret hat. Comments are generally considered second-class content on Stack Exchange sites; they're intended to be used to ask for clarifications or point out things that need improvement, and not stick around permanently. In practice, comments don't always get used in this way, but when they are, they're basically garbage after the requested improvements are made. Here, then, is a hat for cleaning up said garbage.
Close a question using the "dupe-hammer" privilege, or cast a vote to close as dupe that a hammer later finalizes
This is sort of an in-joke for power users, who frequently compare the "dupe-hammer" privilege to Mjölnir, the mythical hammer wielded by Thor, the Norse god of thunder. Another name for Thor is Odinson (he's literally the son of Odin).
It's Always 5 O'Clock Somewhere
Ask a question when it is 5:01 pm on Friday anywhere in the world
There are three meanings in this name. Most obviously, it points out that the hat has to do with the 5 o'clock hour. Also, this is a common phrase associated with the end of the workday and the implication "Why are you still doing job stuff instead of down at the pub?" And third, the presence of the word "somewhere" hints at the fact that the time needs only to be 5:01 in any valid time zone, rather than some specific time zone.
Answer a question that has 0 comments, get 0 comments and score 7
This is pretty self-explanatory in the context of the hat criteria. Each digit corresponds to the number of something that triggers awarding of the hat. The number itself is, of course, famous for being James Bond's MI6 agent ID number. There's no deeper connection on this one.
It's Over 9000!
Get 9000 total views on your questions asked during Winter Bash
Another meme-based hat. (This is a tech start-up, after all.) Dragon Ball fans may remember the original scene.
Win a bounty after another answer first met the auto-awarding criteria
To be clear, the Edward Edwards in question is the British naval officer, not any of the eight other people with that name famous enough to merit a Wikipedia entry. He's most famous today for commanding the ship that was ordered to reclaim HMS Bounty after the famous mutiny. Accordingly, this hat was awarded to users who claimed a bounty that was practically in someone else's grasp.
Historians may complain that this is analogy to SE bounties is imperfect, because the actual Bounty had been destroyed by the time Edwards arrived at Tahiti, or because meeting the auto-award criteria isn't the same as actually having control of the bounty at any point. To which I say, I may have screwed up because I spent a disproportionate amount of time making sure the hat art accurately represented an authentic British naval captain's uniform from the 1790s. And also, look, another distraction, over there!
Receive at least five upvotes and at least five downvotes for the same post on meta
There have been multiple requests for a Winter Bash hat for meta controversy. One, as mentioned earlier (see Fan-hat-ic) was here; another was here. I was especially supportive of the idea because I proposed a badge for essentially the same thing over four years ago.
Getting to the name of the hat was a bit of a process. Early brainstorming—before the design and trigger were associated—came up with the ideas of masks from Peking opera or kabuki theatre as possible hats for this year. Once the trigger was set, it wasn't a far jump from Peking opera to the sometimes-mischievous-but-ultimately-heroic Sun Wukong, who is as famous in east Asia as Superman is in America or Asterix in France.
Ask a meta question tagged bug that scores at least 10 and attracts an answer that also scores at least 10
Computer science may not be as ancient a discipline as most other sciences, but it has its fair share of great people. One of the better-known, and a favorite here at Stack Exchange, is Rear Admiral Grace Hopper. She is a giant in the field for, among other things, inventing the compiler in 1952—a huge step in computing history.
However, this hat is not about that. It's about her less academically significant but more famous literal usage (and subsequent popularization) of the term "debugging" after her teammates found that an error they were encountering was being caused by a moth inside the machine.
(photo from Wikipedia)