If I were Bill Thompson I would do it just to get revenge against Obama for not really supporting him. Some Obama campaign appearances, robocalls, ads, whatever could have allowed Thompson to win.
Just FYI, running a Senate campaign, or any campaign, is a LOT of work. You have to devote your entire life to it... but yes, he should be pissed at Obama for not doing more to help him...I mean, hell, Deeds didn't even want Obama's support and was gonna lose by 20% regardless, Thompson could have won with maybe a tenth of what Deeds got.
But some powerful people in the NYC Democratic circles were backing Bloomberg.
Also, as a side note, Gillibrand and Bloomberg are in different circles.
Two additional notes:
1- I think the White House has shown it will give people stuff in exchange for not running against Kirsten. If Thompson wants a job in the White House or something, all he has to do is run against Gillibrand and let the White House convince him to drop out.
2- Bloomberg and Obama backed Kennedy, not Gillibrand. Gillibrand is part of the Al Sharpton/David Paterson/Hillary wings of NYC politics,