If you regularly use your Game Boy Player(s), apparently Game Boy Interface is far better than the original Nintendo software (I wouldn't know, I bought my player without a disc because that makes it significantly cheaper). There are a few ways to run homebrew on a GC; the two I'm familiar with are using an SSBM exploit, and using a XenoGC mod chip. The former is easier as long as you have SSBM, a softmodded Wii (...yeah, this is getting a bit convoluted - use Wii homebrew to put a save file on a GameCube memory card so you can use it to dump GBA games) and an SDGecko (cheap), the latter requires a bit of soldering and mini-DVDs.
The exploit to get arbitrary code running on your GameCube is called
Home Bros. - that page also links to Game Boy Interface. If you don't have SSBM, there are a few other options in the soft mods forum.
It's worth noting that unlike the Wii, you can't permanently softmod a GameCube, but if you're just wanting to dump a game, it shouldn't be too much hassle.
Once you've got GBI set up, start it up without a cartridge in the slot and it'll automatically boot into a BIOS/ROM dumper. Put your cart in, select Dump ROM, and it'll write it (quite slowly) to an SD gecko in one of the memory card slots.
(Alternatively, the DS route is somewhat easier, as long as you have a slot 1 card; looks like someone else did a far better dumper than mine -
http://gamebrew.org/wiki/GBA_Backup_Tool )
