An assassination in today's media environment would be huge, rivaling 9/11. A similar rally to the flag and doves being quiet would happen. Bush's legacy would get a considerable boost.
Terror and the wars remain the biggest issues in the elections, with even the economy and health care relegated to second tier status. Obama's campaign never gains traction and Clinton cruises to a fairly easy victory over Edwards in the Democratic primaries.
Cheney will have to nominate a VP right away as there will be a scare about further "terrorist" actions and assassinations, as well as health concerns, and Pelosi is next in line until he does.
Just about everything depends on what Cheney wants to do. Considering he never had any interesting in running for President I don't think he would run in 2008, though it's possible he could change his mind after getting a taste. But I'd go with the assumption that he won't.
Any reasonable person nominated by Cheney would likely to be confirmed in this situation. The question is if Cheney wants to pick someone to be a caretaker with him, or if he wants to appoint a raising star and make them heir apparent. Caretaker choices could see people like Gates, Powell, Baker, Kemp, etc. Given that Cheney never threw his weight around much in the primaries in OTL, this seems pretty plausible. There's not anyone great around to appoint as the heir appearent anyways who Cheney likes.
Assuming Cheney gets someone like Powell who isn't interested in running in '08 appointed, it's hard not to see McCain winning the Republican nomination since the issues are going to be more favorable to him. A McCain vs. Clinton close election seems like a very reasonable prediction. If the housing buble can hold off acouple months I'd think McCain would have a slight edge, but a slight edge for Clinton if the housing bubble bursts on schedule.
The Veep picks for such a race would be completely up in the air. McCain might not feel the same need to pander to base with a pick like Palin. Normally Obama might be a good pick, but given his middle name and inexpereince, he would seem a bit unlikely in this situation. For her sake hopefully Clinton would avoid Edwards.