Sometimes, a single-purpose device is the best way to approach a project. Case in point, Adafruit's guide to make a clever little handheld game system that just plays simplified versions of Tetris and Snake.
This little project includes two games, Tetris and Snake, alongside a little paint program. You'll need a bunch of different stuff, but the main component is a small board that handles the brains and the LED matrix that handles the display. Controls are handled by a small little navigation switch.
There are certainly cheaper, easier ways to play either of these games, but this one teaches you a variety of skills as you make it for yourself. Head over to Adafruit for the full guide.
DIY Pocket LED Gamer [Adafruit]