I decided to use the TI beagleboard OMAP evaluation board. It has a nice combination of 600MHz ARM CPU, powerful 3D capable video output, direct access to the LCD signals (but not directly suitable for driving my LCD as I first thought, sadly), and very low power consumption (weeks in standby!). It also has the nice feature of having its filesystem on an SD card, so no moving parts.
Here are some pictures of the beagleboard: