Most US (NTSC) Gamecube games support up to 480p on via component cable (I use an Australian cube with NTSC games via a freeloader disc).
Depending which model of gamecube you have, there may be a separate port on the back for the proprietary component cable (some later models were made without one, and I think in Japan there was a different port altogether). If you have the port, and are lucky, you can get one of these cables from eBay or from Nintendo direct (probably cheaper, if they are still selling them in your country).
If you don't want to spend the money though (or don't have the special port), you could probably get slightly better quality than you have now with a cheap third party S-Video cable (that fits the existing AV port, try eBay).
(I actually use an old high-end-model (but very high-end) 21 inch CRT computer monitor through the component cable and a digital signal transcoder box. The picture quality is fantastic.)
