Update: The Best of MathOverflow is now active!
It seems that there are different kinds of success stories, with different audiences; and we might want to keep them separated. On the one hand, for example, we have mentions of MO in the news or mass media; on the other hand, we have instances of MO-inspired research publications, meaning publications or research achievements that arise principally from an exchange on MO.
These seem to be success stories that are different in character and have different audiences. For example, the mentions of MO in the mass media are generally quite accessible to a wide audience, and could be interesting even for non-mathematicians who are trying to understand what MathOverflow is all about. Accounts of the MO-inspired publications, in contrast, would typically be about a specific piece of mathematics, with a more technical nature.
One way to do it, which seems to be fairly low-maintenance, would simply be to have two questions on meta:
MathOverflow in the news.
MathOverflow-inspired research publications and achievements.
People could post one instance of each type of announcement per entry, and the question would periodically pop to the top of meta, when a new mention was made. A broad interpretation of these two categories would catch most of the announcements we might want to make, and I think it would work well.
Where we should keep track of the stories also depends a bit on whether anybody is willing to invest some time in maintaining this.
What are the possibilities in principle:
On meta. The issue with meta, and I think most agree here and in anycase François G. Dorais recently said so himself, is that first it is not very visible and second if this is not only for internal book-keeping but also for showing to somebody possibly external to MO than meta is perhaps not the best place to direct them (with all its debates and so on). The advantage is it essentiall maintains itself. (This goes both for old and new meta.)
On main. This was tried once or twice, there is however some opposition against this as it is somehow off-topic and with the new meta, most arguments (tex, less discursive,...) in favor of having it on main over meta seem obsolete.
On the blog attached to the site. MO could have a blog hosted by SE (like some sites have) and there this could be "reported" possibly also other things. This might be an alright option if someone or a couple of people volunteer to do this. (As far as I understand, essentially everybody can get writing rights, but one does not have them automatically.) It might not be very visible though, and the format is a bit limited.
Elsewhere. If none of the above, then likely one will need a dedicated site for it, since to host it just one some page might seem a bit confusing. Since MathOverflow exists as an entity it might be reasonable (even independent of this) that it has its own webpage; of course the main (only?) activity of MO is a webpage, still there might be some things to present or to say, which do not fit completely on mathoverflow.net (how is mO organized, its history, its successes and everything else that comes to mind). Of course, there would need to be somebody to maintain this site.
So, in my opinion, the first question to ask is if anybody is willing to invest time in this. If not, the only option is anyway to continue in a self-maintaining way on meta.