I'm not in any way sure he would have been the nominee actually. Incumbent VPs generally always face strong primary challenges, and never would this be more true of Dan Quayle in 1996.
Quayle would be riding on the back of Bush Snr Republicanism; domestically disinterested, nothing more than tactical plays to social issues, foreign affairs competency. Two huge negatives there in a primary campaign, one minor positive. One positive which would be rendered obselete by the settled foreign situation and Quayle's patent inability to carry it over. Plus, the exhaustion factor of 16 years in the White House. After eight years of Bush snr, social conservatives will be angry as hell. Quayle isn't going to convince the center/left and he's got nothing to show the right - I don't think you'll see the party establishment get behind Quayle in any way like they did with Bush in '88 or Gore in 2000. Quayle could point to the (presumably) good economy if he wins the nomination, but the economy doesn't win you primary votes with evangelicals.
He would have people coming at him from the right, left, up, down, sideways. Unless his campaign is operating at political genius-level, he'll probably lose the nomination, yes, probably to Pat Buchanan. (Or at least, someone equally 'fruity') 1996 consequently has a good chance of being the biggest Democratic landslide since Roosevelt.