Mark Thoma says yes (with links to a debate) and I think his analysis is on the mark. Nonetheless he is leaving out one very strong point in favor of his view. The Obama administration has done plenty of interfering with the car companies and also with executive compensation. These episodes make me nervous. Reappointing Bernanke, who is from an opposing party, is a signal that such meddling won't be applied to the Fed and that the Fed will be allowed to regain some of its autonomy vis-a-vis Treasury. Not reappointing Bernanke would make the markets very nervous about the future autonomy of the Fed. (Even if Alex is right more generally about central bank independence, I don't want the current Fed to resemble General Motors or Chrysler.) There's lots of talent in the current White House, but given how much policy has been run from the White House, it would be a bad signal to look to the White House for a Fed pick. Many of the other possible picks seem to be largely untested at a major league level. You can complain about Bernanke all you want but his likely successors probably have the same list of drawbacks that perhaps you are ascribing to him.
So yes, Bernanke should be reappointed.