Gigabyte Brix Pro
(the official measurements have this at 4.5 x 4.3 x 2.4 inches and that's the outer case. The internals probably take up slightly less space then that. By comparison, the measurements for the GameCube is 5.9 × 6.3 × 4.3 inches. Now that I seen the other photos of it, the last number is most likely the height. The Brix Pro is almost half the height of the GameCube. You might actually be able to preserve the original optical drive placement if you are really savy at this kind of thing.

Judging from the measurements, theoretically the guts of this thing could fit inside a game cube. One may need to alter how the case is held together though. There's a good chance the screw mounts might obstruct some things and you might have to fuze the disc drive lid shut and shave off some plastic on the inside that is normally reserved for the disc. But hell, one could make it fit into a GC case easily if they have the modding know how to do it.

It does not appear to have a optical drive. Perhaps one could solve that by modding a GBA Player case and have it mounted on the bottom so you can load in discs Wii style (though you wouldn't have to limit your self to a slot loader type drive). The drive would be connected either via a USB cable buried inside or via mSATA which should also have the cable buried inside. Bonus points if the modder is able to make the thing detachable like the GBA Player was.
EDIT:
After looking more pictures of the Brix Pro, it appears to only be half as tall as the GameCube and is also smaller in size in the other two size dimensions. One could integrate a laptop drive into it and if done right preserve the original door mechanism of the GameCube case. (would definitely need to use a full size case that accepts full size DVDs. Why limit one self to mini discs?

Also, the power supply for this one appears to be external and similar to what laptops currently use, so one would not have to worry about cramming a PSU into the case.

Definitely the best candidate for a project like this.

Photos of Brix Pro:

