How do you build a completely analog "random" system? Building a regular one is easy, building one that might seem random because of how many regular ones are tied together ... but true sources of entropy?
I would imagine that analogue randomness is easier than doing it in a deterministic digital system. Surely there are all sorts of creative methods. Dice or coins in a box? A ball falling through a galton board? Sampling a double-pendulum? Floating particles in a heated liquid?
> 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.