I might be one of the biggest Vice President Evan Bayh supporters on the board, but I think by 2008 he was past his "best-use-by" date. In his favor were that he was a seasoned pol from a conservative midwestern state where his reelection campaign outpaced George W. Bush's margins in 2004; he was a strong and vocal supporter of Hillary who stumped for her and delivered Indiana to her (which I think goes more toward a reason why Obama would select him than a reason why Hillary might); as far as policy goes Bayh served on the Armed Services Committee as well as the Committee on Banking Housing and Urban Affairs (assuming that the Democrats wanted to play up the economic message by demonstrating their chops, it might make Bayh more appealing); finally, during the summer of 2008, Obama was not inevitable the way he was during the autumn and picking a Blue Dog would help him among similar-minded voters (which, it turns out, he didn't really need).
So all of that's going for him. Bayh, in my mind, faced two problems. First, is that he was part of a group of Democrats who were on their way out of power. Within two-and-a-half years the DLC, which embodied the policies of Bayh-like Democrats, would fold. More than this, though, Bayh's reserved style of campaigning was old-fashioned and not in touch with the modern era of politics. Where Obama and Biden (and even Sarah Palin) shone on the trail, Bayh was always robotic and seemed to lack much of a personality. The second problem, which was probably much worse, is that Bayh's opponents had been raising concerns about the confluence his wife's work on corporate boards and his Senate voting record. For a campaign based on "Hope and Change" this would definitely be a liability. (Although, if McCain still goes with Palin then her faults probably overshadow Bayh's failings such as they were).
But assuming that Plouffe's coin flip comes up Bayh instead of Biden, I think the Vice Presidency is going to be a lot less, er, interesting than it has been these past seven years. Bayh will be used as a go between for Obama and moderate Democrats on the PPACA, the Bailouts, and on other more controversial policies. He'll keep quiet for the most part; there won't be nearly as many Biden gaffes or gropes. You won't have him jumping the gun on gay rights the way Biden did, either. Bayh will be a constant presence in Ohio, Michigan, and Pennsylvania during the 2012 elections, and might even be dispatched to Indiana as well if the state appears winnable. I suspect he will endorse all of Obama's agenda as a way of angling for the nomination in 2016 and will point to the 2008 victory in Indiana as being the result of his presence on the ticket.