Willy Wonka and the Charcoal Factory.
This is from a thread by—er—MrStonar titled Idea - Bike powered generator systems. You can probably guess the rest, but the uncropped image is here for all the details. As you can see, the idea is to not directly power things, but to provide a charge to batteries. Though making a game where you sit on a static bike isn't the most fun, I do like the idea of the batteries. It creates an opportunity for people to steal them, undoing your hard work, which is always valuable in the world of Rust.
TBH, we're probably just going to have players rub each other with balloons to create a static charge. We're consulting eminent rubologists on how to make it realistic.