At the time, the Senate was split 55-44-Jim Jeffords, and the House was 232-201-Bernie Sanders, so in effect 55-45 and 232-202 (one vacancy). This means it'd be less complicated than it would get later, but Bush still couldn't cram through anyone he pleased. He'd have to get someone who was unlikely to face a filibuster in the Senate, but be conservative enough for the House majority, and the best way for him to do that is to pick someone out of the Senate, or a well respected Governor.
If he were to go the latter route, Haley Barbour would be a very strong choice, as he had great approval, was well liked in political circles, and his credentials were unquestioned. If picking out of the Senate, working under the assumptions that he wouldn't take from a state with a Democratic Governor or a strong D majority, anyone with very little Senate experience, or someone incredibly controversial, his options would likely com from the following group: Jeff Sessions, Pat Roberts, Sam Brownback, John Cornyn, Kay Bailey Hutchinson, Orrin Hatch, Bob Bennett, Thad Cochran, Kit Bond, Chuck Hagel, Conrad Burns, Mike DeWine, George Voinovich, James Inhofe, Tom Coburn... probably a couple others, but they'd be negligible.
To craft a short list, we're probably looking at Haley Barbour, Pat Roberts, Kay Bailey Hutchinson, Bob Bennett, Thad Cochran, George Voinovich, and Orrin Hatch. If Bush was in the mood for a knock-down, drag-out fight, possibly Donald Rumsfeld out of the Cabinet. I'd put the short odds on Barbour, with Bennett the numbuh 2.