I find the grace period can lend itself to some unscrupulous behavior on popular tags. I've often seen an earlier, minimalist answer be updated within the first 5 minutes, and incorporate something mentioned in a later answer (also in its first 5 minutes, obviously) or expanded upon immensely. It's like they wanted to be "first to post" but didn't want to wait the amount of time it would take them to craft a genuine answer. (While that is the scenario where this applies most often, I don't think my request should only apply when the question is brand new. Especially if it makes the feature more complicated to implement.)
I propose that you shouldn't be willing to hit "Post Your Answer" until you think it is a valid first draft of your answer, and that very first version should be a maintained part of the answer's version history. If I post an answer and then 10 seconds later start editing it, and edit it multiple times, that should be a new grace period, and the initial answer I posted stays intact.
This kind of behavior (especially when it's intentional) is hard to detect because you have to be on the page to see the change happen.
I actually question the value of the grace period. Once you've hit the answer button for the first time, a new grace period should start, and further edits should be tracked separately from the initial submission (with their own 5-minute grace period cycles, same as today). This should eliminate garbage "first post!" answers that are edited later on purpose.