I found a place in my town that does repairs on GameCubes. He said it was a bad motherboard, and he replaced it from a trashed GC that they had in stock. It cost $35...not too bad.
He said to never keep the GC on carpet, and to keep it dusted. That explains the first GC going out, but the replacement GC was only 3 weeks old, on a wood floor and not dusty...so not sure what happened there, but we did get 2 blown motherboards.
If you need repairs, check out places that sell used games and consoles, or that advertise for DVD disc repair. That is how I found this place. They aren't allowed to buy parts from Nintendo, so they buy broken consoles to repair them.