I'm not a huge fan of the presidential pardon power. If I had my druthers, we'd amend the Constitution and get rid of it entirely (Eric -- echoing warnings from the Anti-Federalists -- provides a few examples of how the process can be abused).**
But there are of course legitimate reasons to keep it. For instance, I have a fairly dim view of our state criminal justice systems, and the executive has more structural freedom to correct gross injustices at this level. [UPDATE -- I'm an idiot. As Ugh notes in comments, the President can only pardon for federal offenses. My bad -- I still don't like the power though.] But the power to correct these gross injustices necessarily provides the president with a way to use the pardon power to help cronies or (even worse) to cover up his administration's criminal behavior.
So here's my proposal -- I think we should amend the Constitution to limit the pardon power. Mend it, don't end it. Specifically, the president should be forbidden from granting a pardon throughout the lame duck period. From Day 1 to Year 4, Election Eve, the president would have the power to grant pardons. After that, he or she would be prohibited from doing so until the next inauguration. The idea is provide some sort of political check on the process. If Clinton wanted to pardon Rich, or if Bush wanted to preemptively pardon Cheney, they would have to do so before the voters went to the polls.
It would be even better if we followed Professor Levinson's advice and got rid of the lame duck period altogether. But until that day, limiting the pardon power might help.
[**UPDATE 2: I don't do this very often, but I deleted a link and reference to a post which I (in my haste) thought responded to the merits of Eric's post, but really didn't. I shouldn't have included it, and I know that I'm outside the window for deleting this. But it was driving me crazy, so I've deleted it. I apologize, but I also think it's tangential to the substance of the post.]