A Vice Presidential choice should always be exciting, and I just don't think Joe Biden is all that exciting. He may be a very nice man, a good man, but I don't see him as a future President.
A Clinton/Obama ticket would have so much better, but that is my view. It would have given Mrs. Clinton at least eight years as the first woman President and then given Barack Obama, experienced, and maybe a little grey, and most likely still popular, the future.
But it may just work the other way in 2016. Anyway, back to the question.
I would have tried to get Olympia Snowe or Susan Collins to join the ticket. But I guess the best choice would have been Tim Kaine (just to make the slogan "ObamaKaine" which has a catchy ring to it), since he has nowhere really to go now (Democrats already hold the two Senate seats and he is term-limited as Governor).
I agree Kaine would be an appealing candidate.
But in recent years it seems that Vice Presidency is not really longer a place for a heir-appearent. Not only because from all sitting veeps since Van Buren only Bush Sr. had been elected (not to mention of course those, who succeeded the office in midterm). Cheney made this clear at very start he won't seek presidency in 2008. Biden, regarding his age, probably won't be a candidate in 2016 (74 is pretty much even for a politician
) and his only chance to became President is succession. Objectively, he's a great person heartbet away.
Now, if I were Barack Obama I would probably choose Kaine for strictly electoral purpouses. I would never chose Bayh. If I feel confident enough I'd love to go outside the political template. For example Vice President Cantwell is my wet dream, but unfortunatelly I would not be possible
