Many posts suggest doing away with the cable by putting the motor on the elevator car; but this overlooks the fact that the elevator needs to be connected to a counterweight for efficiency reasons.
However, here's a thought: you put motors on the elevator *and* the counterweight. As the elevator goes up, the counterweight goes down and uses its motors as generators to partly power the elevator's motors. And vice versa.
Sure, you're not going to break even due to electrical losses; but it'll be a damn sight better than no counterweight.