I don't know hardly anything about Congresswoman Blackburn. Judging from the front page of her website it seems she and I share some of the same priorities. But to be honest I think it would be an extreme longshot to run two Representatives of the House on the same ticket(even if they were both exemplary human beings). Enter someone with the 'star power' as Kevmo put it like a Secretary Rice and you could be going places.
As for the scenario of Bush dumping Cheney to give a Republican a good platform to run for President I don't see that happening. If Bush did decide to dump Cheney I think the person he would most likely replace him with would be Condi Rice and she already has a platform should she decide to run.
I agree with you for the most part, although I frankly think the bias against House members is something we'd be well served to leave to history, since it may prove to be a counter-productive limitation that deprives us of some exemplary candidates.
I'd be fine with Condi, but as yet have seen no indication that she's willing to run. I also haven't personally examined her enough to know how deep her conservatism run, and I'd be concerned that we might be setting ourselves up for a party split later on if she runs for the top job. However, there's no denying she would be a very powerful part of a ticket, and I certainly wouldn't object.