He should have selected John Danforth as many expected him to. Not Darth Cheney.
Actually, most people thought he would chose either Tom Ridge or Elizabeth Dole.
They would have been good choices. I'm not sure whether Ridge would have helped Bush carry Pennsylvania but he would certainly have bought something to the ticket that Bush lacked. Dole would have been groundbreaking being a woman, as well as in keeping with official Republican policy since 1976 that a Bush or a Dole must be on every ticket nominated.
I wonder if Elizabeth Dole would have been ambitious enough to consider running for President in 2008, despite being 72 years-old? It hasn't stopped McCain. And would Ridge have opted not to run for re-election with Bush in 2004? After all, he retired from the administration in real life at the beginning of the second term. John Danforth would certainly have been a good choice; experienced, more moderate and from a critical bellweather state. What is interesting to hypothesise is whether he would have publicly broken with the administration and the GOP as he has more or less now? Probably like Colin Powell, he would have retired after one term in 2004 and just distanced himself from Bush. So if Bush did select Danforth or Ridge and they retired after one term, who would he have chosen in 2004? I sense Bill Frist would have been high up on the list.