One of the presents I received for Christmas last year was the Arduboy, which is a fairly fun Arduino-based handheld. However, without any immediate inspiration, the first two projects I created treat the Arduboy as a peripheral rather than a game console.
The first project is the Arduboy remote, which assigns keyboard keys to the Arduboy so it can control some operations on a connected host device (PC or otherwise). The default keys are arrows, space and enter, though the firmware can be edited to change these.
Additionally, the Arduboy will listen on the serial port at 19200 baud. Any text sent to the Arduboy will be displayed on its screen. If a form-feed character is sent, it will clear the screen.
The second project is a simple Joystick firmware for the Arduboy, using the two-button variant of my Basic Joystick Arduino Library. This allows you to use the Arduboy to play games on PC, albeit with a very limited button set. In practice, this is only particularly useful for emulators and perhaps the odd indie game if the button requirements are humble enough.