George Bush ... will want the next Republican presidential nominee to continue the U.S. effort in, and commitment to, Iraq. To be a candidate who will continue his policy.
The best way to assure that is to have the Veep running, since Republicans still respect primogeniture in Noonan's words. The question becomes who would replace Cheney?
I've discussed Condi Rice as a candidate before. She'd have an edge in the general election as the first black woman with a chance to win. Iraq is part of her legacy too, so she won't bail out on it. A presidential endorsement would help overcome her weakness with the conservatives on social issues. OTOH, she's never won an election. She might not have enough popular support to be respected as a contender. And she'd have to be approved by the Senate, which includes several guys who want that job themselves.
That brings me to John McCain. He's got the support among moderates and the media to be considered worthy of the job. Senators are usually good at supporting each other so he'd win that vote. He's been a clear and consistent hawk on Iraq, so Bush wouldn't have any doubts on that front. The main objection to him would be his personal conflicts with Bush and other parts of the Republican party. Not insuperable, but McCain probably won't get picked if they can find someone else who'd have a good shot at winning in 2008.
So who else would it be? Jeb Bush wouldn't need the Veep boost to win as a big-state governor, but I don't think many Republicans are willing to put up with three presidents from one family. Rudy Giuliani might work. Any other nominees?