I can't see it. It's a crap job -- the only reason to want it is to boost your resume for a later run at POTUS, but Thompson would be too old to run in eight years. (Cheney's an exception, but as far as I can tell, he was asked to take the job.)
Actually, Cheney has had a great deal of power and influence..much more than most realize. He and Bush have reshaped, redefined the role of the VP, and the relationship of the two positions, which of course was easy to do once Cheney annoucned he had no inten tions of ever running himself. Imagine is Rudy picks Fred, or Newt ( my original choice...see my tagline) and announces that he will be focusing mainly on the War, and our foreign policy and will be delegating much oversight for domestic issues to the Veep..