A rear-view mirror would be a reasonable use of the UPad, especially in single-player or online, but I don't think it's the best choice as a "standard", especially due to the asymmetry issue.
So here's some ideas that come to mind. First, for "regular" matches, the best use would have to be as an extra screen (for up to 5 player local multiplayer). For four-player multiplayer with the UPad not used, the best use of the UPad would have to be for the map, positions, etc, to keep them off the main screen.
But it's the "special" modes that are more interesting in terms of ideas.
Idea 1: U and Me mode - Two player battle mode (playable online), where one player controls the kart and is on the TV, while the other player uses items and can either use gyro control to have a VR display type setup or analog control for less effort. Could even include special items designed for this mode, such as a manual-control shell. Think of it as an evolved version of Double Dash.
Note: simple "shoot shells like you shoot ninja stars in Nintendo Land" is a bad idea - it's gimmicky and would involve using that format for no other reason than that you can. As Jim Sterling has pointed out, gimmicky use of touch and motion feels like they're still tech demos. Nintendo Land is effectively a tech demo, so it's fine, but otherwise, don't include it unless it improves the game.
Idea 2: Mapmaster mode - up to four players on either battle maps or regular races, while a player using the UPad controls the map, rather than a racer. Their goal? To help the computer racers/battlers to beat the real players. By having the computer able to be set to three difficulties, the difficulty for the mapmaster can be adjusted (inversely - better computer makes it easier for the mapmaster).
Idea 3: Hidden Obstacle mode - Single-player mode in which the player must keep track of both the TV and UPad screens, as different obstacles will show up on each screen. It's part of the kind of Challenge mode that was found in Mario Kart DS.