Since the smart mirror project is essentially an embedded system, the easy choice for the hardware was a Raspberry Pi. They're cheap, well documented, and more than powerful enough for my needs (plus I wanted an excuse to buy one).
I also purchased a 5" TFT to use with it (<£15 on Amazon) which has a composite video input.
Having set everything up, I'm thinking of switching out the TFT for something that uses HDMI as an input. The resolution of the screen (800 x 480) is a lot to ask of composite video, and the result is a slightly flickery image.
Wifi was super easy to set up, generic drivers for the dongle I bought got installed straight away. Performance seems to be lacking but that's a problem that can probably be solved with a driver update. The (Poundland) bluetooth dongle I picked up doesn't seem to work OOB, so that will require a bit of work to investigate.
Next steps: installing Processing, Git and SSH to enable me to push code to the Pi and run it.