A Nintendo patent from March 2006 shows that the company originally intended a Wii-mote style device for the GameCube. A patent drawing shows how the Wii Remote could link with the GC.
A Wavebird-like adaptor would be attached to the GameCube console, which would interface with the wand remote. There appears to be two separate sensors points on the television.
The wand remote in the patent drawings looks surprisingly similar to the finalised Wii-mote.
There is also a patent drawing for a tennis game — which includes a drawing of Mario. Early shades of Wii Tennis, no doubt. So somewhere along the line, Nintendo decided to turn this add-on into the impetus for the console we now call the Wii.
Nintendo Patent Reveals [Siliconera]