A lot of the discussions so far have revolved around FAQ content (what it should contain, how it should be answered...). Amanda already got the ball rolling with a list of possible topics: What should our FAQ contain?
There is, however, a bit of confusion on how the FAQ should be updated. As it turns out, we do not have any access to the actual site FAQ (linked from the top bar), which is meant to be identical across all sites, except for the 'What kind of questions can I ask here?' part.
From a thorough reading of meta.SO and discussions with SE moderators: the proper way of building our site-specific FAQ is by tagging well-defined FAQ question/answer with the faq tag here on meta.
The process should go like:
List potential FAQ topics (already taken care of by Amanda here), or simply suggest a FAQ item directly as a meta question.
Formulate the question as close to what the final FAQ item should look like (e.g. "Where can I find resources to learn Japanese?" or "Are purely cultural questions allowed on JLU?"). Optionally explain why this FAQ item would be useful and what its answer should cover. Jeopardy style questions are acceptable at this point (where you post both the question and the answer).
Tag it with faq-proposed and make any initial answers CW. The moderators will mark the meta-question CW after it is posted.
When people have weighed in with their answer, somebody should build a consensus answer. This answer should appropriately sum-up the consensus, in a clear, concise way. It should avoid unnecessary digressions and comments that will have been addressed beforehand.
Edit the question to give its final, concise FAQ format (removing all peripheral/extraneous meta-FAQ discussion). Tag it with faq Leaving comments that illustrate any debates/discussion on the post is acceptable to prevent future questions, if they don't have valid room in the answer.
Mark the consensus answer as the 'accepted answer', so it appears on top. People who posted during the consensus-building phase may want to delete their intermediate answer. Moderators may delete similar extraneous answers.
Voilà, we have a shiny new FAQ entry that new users can be referred to.
Most importantly, all faq entries are linked from the site's permanent FAQ page.
What this means for now:
We should keep brainstorming about new FAQ entries (and either post suggestions over there or directly in meta).
All previously asked questions that are good FAQ candidates should be retagged faq-proposed (currently, most of them use faq-topics, which should be merged with faq-proposed).
Questions that have been discussed at length, should start working through a consensus answer (see for example: Should uniquely cultural questions be acceptable?).
If no consensus can be built, then it probably shouldn't be a FAQ entry and should be marked faq-declined.
Sometimes faq questions already exist on meta.stackoverflow. It is sufficient and permissible to cross-post those questions and their answers, but it is preferred to refer people to those questions as well, where possible. It is considered "proper procedure" however, to post any faq questions you may have here. There's no reason to cross post them (or to start the process) on meta.stackoverflow.
Additional suggestion: we could build our own meta-question with a list of all current faq questions and a one-line sum-up of the consensus (linking to the full detailed answer), for easier perusal. Some SE sites do that and I think it's a good idea (but more work). Anyway, this is not the priority.