So, the "butterfly effect" refers to the idea of chaos theory - "sensitive dependence on initial conditions." That made me think of weather systems for a while, but I don't think I could implement anything along those lines in the time alotted.

The other canonical example of a chaotic system is n-body gravity. I thought about some stuff with planning routes through the solar system, but even that I think would be too complicated.

There's an old old game concept that's been re-used a bunch of times, where you have two spaceships and they have to fire projectiles at each-other, that are affected by gravity. I think a variation on that might be interesting and on-theme, so I'm gonna go with that; with not just the projectiles affected by gravity, but with the ships and a bunch of asteroids all with simulated gravity - so firing your missile or moving your ship will have an effect on the position of the asteroids, which affects how you have to aim your shots and move your ship, and you have to try to predict what effects your actions will have in that chaotic system.