My brother's friends treat them awfully. They've broken a total of SIX, yes SIX Gamecube controllers, gotten the reset button stuck, AND broken the first controller port. I just fixed the first controller port and the reset button, but I had to throw out all the bad controllers.
Anyway, I have the solution. To fix it, all you will need is a soldering iron, solder, tape, and a 4.5mm gamebit screwdriver to open the Gamecube. The tape can just be scotch tape, the soldering should have a fine tip (you can buy one at Radioshack), the solder you can get at Radioshack, and the gamebit here:
http://www.jandaman.com/games.mvc?p=gamebit45&Category_code=GC. You can also do a google search for gamebit 4.5mm if you want to try for a better price. Some tool shops even have it. NOTE: Check inside the Gamecube to see which problems you have BEFORE buying supplies, as you might not need all of the supplies. The only tool I assure you that you will need is the Gamebit screwdrier. Right, so using the gamebit screwdriver, you unscrew the four corner screws on the Gamecube (look on the bottom, each corner should have a hole. The screws are at the bottom of a hole). Then flip over the Gamecube (BE SURE TO HOLD BOTH THE TOP AND THE BOTTOM OR IT WILL FALL APART). Then, take the top off. Be sure not to lose any of the screws, or the top. Then, unclip the grey front panel. Near the center, there should be a number of metal lines which go off into a cable. Now, there will be between 1 and 3 things wrong here. The first is that one of the metal connectors hooked up to the cable will be burnt, and broken. This is where the soldering iron and solder comes in. Now, this is the difficult part. Your goal is to solder the broken connector back together. Plug in the soldering iron, and wait for it to heat up. Then, bridge the gap in the connector with solder (melt the solder with the iron at the gap). DO NOT BRIDGE THIS CONNECTOR WITH ANY OTHER CONNECTOR. IF ANY CONNECTORS ARE BRIDGED YOUR GAMECUBE WILL BECOME SERIOUSLY DAMAGED WHEN POWERED UP. BE SURE THE GAP IN THE CONNECTOR IS THE ONLY THING BRIDGED, AND THAT NO TWO CONNECTORS ARE BRIDGED. Then, hold the soldering iron to the solder for about 10-30 seconds, until the solder starts to "flow" (which is when it is really liquidy). Then, stop. Remove the soldering iron from the place you are soldering, and unplug it. The solder will dry. REMEMBER TO MAKE SURE ONLY THE GAP IN THE CONNECTOR IS BRIDGED! IF ANY CONNECTORS ARE BRIDGED, IT WILL SEVERLY DAMAGE YOUR GAMECUBE. Now, that was the hard fix. The next fix is far simpler. This is the fix where you need the scotch tape. Ok, look to where on the board the cable is going. Right where the actual cable (the cable itself, not the plastic connector on the end) meets the plastic cable, the cable might be pulled out. Push the cable into the plastic connector. There should be a good place along the cable to push from. It will crease the cable, but it doesn't matter. Now tape the cable into the plastic connector. The final thing is to check whether the plastic connector is plugged in all the way. If it isn't, plug it in all the way.
After all of that, close up your gamecube in the manner you opened it. Stick the top back on, and check to see that all the buttons of the Gamecube are in the proper position. Also make sure they work properly. Then, flip over the Gamecube with a hand on either side of it (so it doesn't fall apart), and replace the screws in the holes, and screw them in. Now, test your Gamecube for an EXTENDED period of time. One of four things should happen. If your Gamecube boots, and you find that the first controller port works, then congratulate yourself. You've fixed it. If the first controller port still doesn't work, or cuts out after a while, try opening up your Gamecube again, and firming the plastic connector and/or the cable, and/or holding the soldering iron on the solder a while longer (10-20 seconds). Be sure not to burn the board with the iron. Keep trying until it works. If the cube doesn't boot, you are SCREWED. Check to make sure no connectors are bridged, and that everything is in order. Hopefully, its jsut the top on wrong, but you could've just burnt out your cube. Ok, the last possibility is that the first controller cuts out after a very long time. If this is so, try unplugging it and replugging it. If you are worried, carry out the operation until it works.
Anyway, that's the best I can do. If none of that works, that you are out of luck. I give no guarentees, as I have only found this to work on one system. I also don't guarantee you won't damage your Gamecube more, or break it. I tried to reduce that with warnings, but hey, people are people and we make mistakes.
Good luck!