> How do I make a mechanical thing happen at a random time with a lava lamp?

Use a heat lamp interrupted by the lava globules to activate a wax motor. If you get the angles right you can probably do this with the same light that runs the lamp itself, or you can put another lamp at a 90° angle (but will have to adjust the main lamp to keep the total heat at the level you want).

Or do similar with muscle wire; more temperature needed to trigger the actuator, but you can get them much smaller so the total heat can be smaller, if you run e.g. a collimated infrared laser as your heater across the lamp.