Nothing would change if Cheney was the nominee in 2008. Bush was unpopular by that time and voters would have seen him as a continuation of Bush's administration.

Obama would win by a larger margin than in OTL. Cheney was disliked by many for basically being the driving politician behind the Iraq War and the false accusations of them having WMDs.

On second thought, I think Obama would win Missouri and Montana.

By the way, I used YAPms to make this map if anyone is curious.