Yes please!
And since a picture is worth a thousand words (or so they say), here's a few suggestions how that could work on the main site. Please note I've included all the suggestions I could think of in a single image, and might look slightly too cluttered:

Wonders of Chrome Developer Tools, but the point is, as far as design is concerned, it is all already there, I haven't used a single new CSS class to produce this HTML that isn't already a part of standard Stack Exchange design. Let's describe a few things:
- Main site banner, as can be seen on early betas to invite members to participate on meta, is used to expose currently selected featured tag. The first link would either link to the new questions tab (see below), or same as any other tag links work already. The Learn more... would link to the meta thread that is selected for selection of these tags, or maybe a new help system subpage describing the selection process.
- New questions tab names the featured tag and lists all the questions tagged with it. There could be an indicator icon showing how many such tagged questions are currently on the site, same as the bounty indicator tab icon does. Tags themselves are colored as featured, same as they are on meta pages.
- Community Bulletin on the right hand side lists currently featured tag and links same as the site top Featured Tag Banner does. If there wouldn't be space, it could alternatively be added to the list of tag filtering tools below the Community Bulletin box.
As mentioned previously, using all these options at the same time might be a bit much, but I'd leave that decision to Stack Exchange designers. Alternatively, featured tags could be colored using something else than the intimidating red.
Perhaps the accept green:

Or even the banner yellow color scheme:

As per @Undo's suggestion, but it would require adding a new CSS class to already existing ones. There is however an already existing required-tag
class (the dark gray tag in default color scheme from meta) that I'm not sure is actually ever used on the main site, so I guess that one could simply be edited to use new colors.
How could this selection process work?
Moderators would have to have a new option in their tools to select a featured tag, not unlike they have the ability to edit tags themselves. It would probably be best limiting this selection to a single tag, each new selection overwriting the previous one (if it at all exists). Ability to automate their validity can be optional, moderators are expected to be active enough on the site anyway, and reminded by other users in meta or in chats if they forgot to swap featured tags for the week, or however the site has decided to handle them.
Additionally, moderators would be responsible to write a few new pages, namely the description of the selection process itself in the help center (new subpage), and then running this selection process through meta threads, where community can chip in with their ideas. Of course, all controlled to be kept within the previously agreed on scope of a particular Stack Exchange website.
I know it might not be popular here on mSO to say anything is easy, but I'll say it anyway - it ought to be easy enough to do. Most of the required stuff is already there, and this would be merely extending their capability. It would require a bit of thought though, how to do it so it can also be worked with as simple as possible. My take? It should be perfectly doable, and could provide for a great new feature, in particular to young, yet evolving Stack Exchange websites.
This however, I'll repeat it from the proposal, probably wouldn't be the best idea for larger Stack Exchange websites, such as Stack Overflow, Superuser, Server Fault, e.t.c. Then again, they don't really have problems controlling their scope or fill their space with new questions, do they?