Some ways in which I've seen luck used in a game:
- Determining injuries in scenarios like car crashes, large explosions and collapsing buildings - roll under your luck score or you end up with something Bad happening.
- Determining miscellania in games where the players are already "established". Is it likely my character has any aspirin lying around in his flat? Roll luck. Does my character have enough loose change to buy a pack of cigarettes? Roll luck. Does my character's servant know how to read and write? Roll luck.
- Allowing players to get out of a bad scenario if they make a successful luck score, with the caveat that if they fail things will get worse. Your character falls off a horse - you can roll luck to see if you avoid a broken leg, but if you fail you get a broken neck. (Or you can just take the broken leg and not gamble.)
I like all of these, because they do away with having the GM have to adjudicate in many situations. They also inject randomness into the game, which is always more fun.