As Stitches mentioned, the Gamecube is a *very* depreciated system. If you go for a Wii portable, you can still play Gamecube games on original hardware, as the Wii is just a beefed up Gamecube, hardware wise (loose Shank quote). You can trim a Wii significantly smaller, it will generate far less heat and consume far less power. The documentation for Wii trimming has grown exponentially over the past couple of years, and companies like
4LayerTech and
Electron Shepherd have designed products that make Wii portablizing a very beginner-friendly process. Even if you have a Gamecube laying around, you'd still be spending hundreds of dollars to transform it into a portable, and designing all the portablizing solutions (like power, audio, controller, etc.) will be *a lot* harder than going with some of the premade solutions by the aforementioned companies. You can get a 4-layer Wii with a busted disc drive for fairly cheap on eBay.
I would suggest reading up on the following threads / guides:
Wii Trimming Guide
Wii Trimming Manual
Gboy Rev 3 Assembly Manual
Shank also does a great job of explaining why Wii portables are vastly superior to GameCube portables in his WiiVision Video: