I think everyone can agree that there is really nothing all that compelling about the President's Cup. There isn't the passion of the Ryder Cup, the US has dominated, and not even the captain's seem interested in making the the match-ups exciting for TV. So here is what I propose, and for the sake of argument, let's keep it a US vs. the non-Euro world.
1) Introduce more strategy into the play. Every hole should have 2-3 sets of tees and as long as you have the honor, you pick the tee.
2) Let the players pick their singles match-up playground style. Pick a name out of a hat and that guy picks his opponent on the spot