Preamble. When thinking about situations like this, it's tempting to fall back on traditional ideas. I realized earlier today that everything suggested so far would also work if MathOverflow was a giant stack of loose leaf paper instead. Though this is a characteristic of excellent ideas that are both simple and versatile, it does suggest that we might not be considering the full extent of our options. I want to encourage users to think outside the box while considering this issue. The suggestion below is a small step in this direction and I my hope is that our collective creativity will lead to a solution which is as innovative and practical as MathOverflow itself.
The idea that I will explain is motivated by several issues. One issue is the problem of deleted content that I raised in another answer. Similar issues are brought up by edits, revisions, later answers, etc. Another is that, although I expressed doubt about this, the fact that some users would prefer that only some of MathOverflow's content should be preserved suggests that this is a desirable feature. Finally, I realized during in my research on archiving agencies that they could find the total volume and regular flow of MathOverflow problematic.
This suggests that MathOverflow content could be archived only on demand. That is, there would be a MathOverflow Archive which stores a static copy of a question or answer in its current state when, and only when, a user requests it. Thus, if you want to cite something on MathOverflow, you make such a request for the content you want to cite, that content gets archived and your citation will always point to a static copy of that content.
This basically solves the deletion and editing issue since I doubt anyone would make such a request for deleted content and static copies remain as they are even if the original gets edited or modified in other ways. This also solves the selectivity problem since (presumably) only good content would be requested in this way. Finally, this also solves the volume problem since much less content would be archived by request.
This kind of archiving mechanism might also serve as support for other possibly desirable features such as an option to have the content peer reviewed (preferably in an open way using the Selected Papers Network, for example) and, if the content is approved, it would get stored in a designated "peer-reviewed" part of the MathOverflow Archive (along with the referee report). Such enhancements would definitely give more credibility to MathOverflow as a micropublishing platform.
Scott pointed out that a defect of this approach is that few would bother making such requests. A potential solution is to have posts automatically archived as soon as someone presses the cite button to get a bibtex/amsref entry and have the urls there point to the proper place on archive.mathoverflow.net instead of the original.