Here is a past discussion on pseudonyms from MO.1 days: http://tea.mathoverflow.net/discussion/335/1/why-not-real-names/. I think most of the arguments (pro or con) would still apply to MO.2. (This find is based on a quick google search, but I think there may have been more discussions. Please feel free to edit this answer if you find more.) 

Women have often reported very negative experiences posting under their real names, for a variety of reasons. This is touched upon here: http://tea.mathoverflow.net/discussion/985/. This alone seems to be a serious trump card in favor of allowing pseudonyms. And it's not just women: many people have had very unpleasant experiences posting under real names (again for various reasons); I believe their needs should be respected. 

(I say all this despite the fact that I *much prefer* interacting with someone who uses his/her real name: I might have the pleasure of meeting them some day, or I might want to cite them, or maybe even work with them on something, and so on.) 

User alvarezpaiva (whom I know btw as "JC" or Juan-Carlos, from grad school days) has made some very good points in support of *strongly encouraging* the use of real names, but *disallowing* pseudonyms I really can't get behind. 

Last point: my own impression is that the rise of questions unsuited for MO is more accurately explained by the merger into the StackExchange network than it is by usage of pseudonyms. Some interesting reading material, with graphs, can be found here: http://meta.mathoverflow.net/questions/302/has-the-switch-to-se-2-0-increased-the-rate-of-low-level-inappropriate-questio