Picking up on Howard Fineman's speculations, I've been thinking a little about this for a few weeks at least.
Bayh and Vilsack seem obvious choices to me. I don't know what to think of Richardson as VP. While he's certainly knowlegeable on foreign affairs he still seems like an odd duck to me. Too easy to imagine him making a gaffe or drawing attention to himself in an unhelpful way. Think of Bob Graham and the diaries.
I hadn't thought of Salazar, the Senator from Colorado, or Bredesen, the governor of Tennessee. And I don't know enough about them to have an opinion on them yet. Both come from states that could be up for grabs, although Tennessee may still be too far away.
How about Joe Biden? He wouldn't bring anything electorally to the position, but he would bring a reputation for foreign policy "seriousness" and experience more broadly. I think the pundit class likes him, and that might not be too bad. If you can't beat 'em, join 'em. But Biden talks too much, and like Richardson, might not be inclined to keep things under wraps as a suitable VP.
How about if Obama is the nominee?
I'm inclined to think that someone like Bill Bradley might fit Obama pretty well. Someone more experienced, a somewhat "reformer" reputation, someone to compliment Obama's campaign against the politics of usual. That was sort of Bradley's schtick in 2000 and if I'm not mistaken, he wrote a book about it. Bradley would bring a "substance over style" impression and I don't think he'd upstage Obama but would nonetheless be a well-regarded pick. A sort of "elder statesman" choice, not too far removed from politics or too old, but despite his experience, different.