YG300 pico projector
What I'm not sure about yet is how this will be implemented but I've been thinking about building a low resolution matrix of lasers and doing something with persistence of vision as you move it.
Which may of course be completely impractical.
With a Raspberry Pi connected video playback of 4:3 SD content with omxplayer is actually perfectly acceptable for your £30. It's very dim, you have to use it in total darkness but turn the lights off and it's watchable. It'll also play off USB/SD.
These can take an optional battery pack and there's a space behind the mirror for this. I did a little poking around with a meter and there are three unused connections on the top right of the board.
B - Ground
R - Battery positive, it expects a 2S Lipo pack so ~7.2-8.4v
Y - Supply positive which by default is 12V
I assume there's a balance charger built into the battery pack as there seems to be no charging circuit on the board. It runs quite happily off these connections and provides a little battery meter on the menu screens.
Using these connections seems perfectly reasonable although I can see myself making them external rather than squeezing them into the case.
I'd like to run this off a Raspberry Pi Zero, although equally something like an ESP32 can be wired up very easily to generate composite video. At 32x240 it really doesn't need to connect over HDMI.
Space inside is very limited but it would not be impossible to 3D print a replacement case that spaces the elements out like they are in in the original.
It could perhaps make for an interesting "pepper's ghost" setup that emulates a sci-fi hologram projector, just so long as it's in the dark.