The 'make game' button is a bit of a holy grail, but I could imagine that since many games ideas are highly parameterised you could have a succession of games effectively bred from previous versions by 'mutating' those parameters as you go. Not sure if this could include as large a change as switching genre, without an awful lot of work, but perhaps if the 'engine' was a mixture of types then you could just blend between them as the parameters change. This is along the lines of genetic algorithms and 'breeding' of systems like the Biomorphs in The Blind Watchmaker.
If you meant 'how do I implement this' then that's another question. Probably best to start tinkering in Unity, perhaps with some existing game examples. Move as much of the 'magic numbers' and 'adjustable properties' out to some central place so they can be saved and loaded, then you have a basis to start 'mutating' the games behaviour, or spawning new games (new files) that can be played too
Hey Sarainia! I actually do this kind of thing during my day job as an AI researcher. I recently open-sourced a little Unity example of a system that generates very simple two-player games: https://github.com/possibilityspace/bluecap and I made a video to go along with it (although it's very long and includes a load of other, unrelated things you can skip:
We've had a few PROCJAM entries in the past that randomised rules too - there's one about using cards to influence the outcome of a battle between two AIs, I forget the name of it now though...
