You are right, it is probably a overheating problem. And if you let it continue it will wreck your Gamecube PCB chip because no chips can take excessive heat. The early first generation Gamecube models had a problem with overhheating. The Gamecube fan is small and doesn't really move much air. So if it is not working effectively, it can cause damage to your components. So always keep the vents clean and unobstructed.
Make sure that wherever you have your Gamecube, the vents are not blocked from air flow. Never block the vents. If vents are clear and fan is running well, another solution used effectively is to hook up a small external fan with a higher cfm to blow into the Gamecube. You can buy those here at the shop. Personally, I recommend putting them on all Gamecubes. They snap right on so there's no worry about installation.
If all else fails, you will have to replace the motherboard or send it in for repairs..