I have a composite output cable for my Gamecube, but it would be nice if it supported a higher quality output, such as S-Video or Component. Does the Gamecube have output support for these outputs?
Some Gamecubes (DOL-001) support Component out with the proper (discontinued) cable.
Most will support the (also discontinued) S-Video Cable
And all should support Coax and Composite.
Note that most US (NTSC) Gamecube games support up to 480p via component cable (I use an Australian cube with NTSC games via a freeloader disc).
As McKay says, 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, you can get one of these cables from eBay (US$50-100). There may even be a chance of still getting one from Nintendo direct in some countries (unlikely, but more like $35, so might be worth the call - I bought mine long after the Gamecube was no longer being sold).
If you don't want to spend the money though (or don't have the special port), the next best thing is a cheap third party S-Video cable (that fits the existing AV port, try eBay - less than $10 shipped).
(I actually use an old (but very high-end) 21 inch CRT computer monitor (set to RGB colour mode) through the component cable and a digital signal transcoder box. The picture quality is fantastic.)