A handful of sites have conducted a variation of the "weekly topic challenge" idea. I, myself, have become quite fond of them since they do seem to be effective at increasing asking rate and can be a lot of fun. I've personally initiated challenges on several sites (most recently on History and Philosophy) and have worked out a simple process.
Step 1: Ask for topic ideas.
On the per-site meta, write up a question asking for ideas. Typically, people will suggest topics that they feel are underrepresented on their site. But often unloved topics don't get as much of a response as subjects the community already asks about. Instead, it can often help to find a tag that is particularly on people's minds at the moment. So, for instance, just before Memorial Day (in the US) is a good time to suggest BBQ questions on Cooking. You could suggest a taffy week instead, but be prepared to be underwhelmed.
Feel free to copy the text from one of my challenges, but it's best to write in your own voice and target the audience of the site you are on. Tag your question discussion, challenge, and ask a moderator to make it featured. Ask for one suggestion per answer. After a week or so, you should have a small stable of topics neatly organized by community vote. At that point you are ready for:
Step 2: Post a challenge.
Pick a day of the week you'll have some time to write up challenges and tally results. Then pick one of the proposed topics and post a question on meta challenging users to ask questions on that specific subject. If you already have an appropriate tag, remind folks to use it. If no single tag fits, suggest a tag set (like batman and nietzsche). Don't worry too much if the tags won't be useful in the future; you can always retag at the end of the challenge.
I like to give people a full week from Friday to Friday. That way people have the weekend to ponder the challenge and can start asking when they get time on Monday. It's often useful to say how many questions the site currently has on the topic and link back to the answer from step 1 that prompted the challenge. At the end of the week, proceed to:
Profit!!!! Tally the results.
After the challenge is over, it's nice to write up a simple answer that enumerates the questions generated. There's a danger here, however. If you only get one or two questions (or zero!), you might be tempted to call the challenge a failure. But think of it more like playing poker: you have to lose a few hands in order to win others. Who can say why, but some weeks and some topics strike gold producing substantially more activity than others. If you give up after a week or two, you won't be able to hit the lucky topic that brings in many questions.
Step 4: Repeat until you run out of topics.
I like to start challenges back to back. That way people contemplate the upcoming challenge while the previous one is still on their minds. All good things must come to an end and a topic challenge is no exception. A good time to stop is when you run out of topics from step 1. If you start to lose steam before that, don't worry about it. But you might leave a note asking someone else to volunteer to take over.
If you do run a challenge, please drop me a line (see my profile) or reply to this meta question. I'd like to keep tabs on all of our weekly topic challenges. Thank you in advance.