When it comes to DIY game consoles, the Raspberry Pi is king, but it's certainly not the only player in the game. BeagleBone is another cheap Linux board, and BES is an operating system that will cram in a bunch of emulators for an easy-to-make emulation machine.
When we last mentioned this project, it was SNES-only, but over the years they have added in support for GameBoy, GameBoy Color, GameBoy Advance and NES as well. Like similar options on the Raspberry Pi, BES is a full operating system, so you'll simply need to download the SD card image, copy it over, then follow the on-screen instructions to get it up and running. You'll find everything you need to make one for yourself over on the BES page.