Some new images and details on Nintendo's patent for the Wiimote have surfaced on the net and they show that the original idea was planned for the GameCube. The Wiimote was to attach to the console via an adapter, much like the WaveBird's. But, as you all know, the GameCube ended up going the way of the dodo, and so the Wii was born.
The image to the right shows Mario playing tennis, which is where Nintendo originally got the idea for Wii Sports, we assume, but there's more. Past the break, you can see more diagrams showing how the Wiimote would work with the GameCube, if you're into that kind of thing.