The CPU and GPU connect to the motherboard through hundreds of balls of solder in between the two components.
As the 360's motherboard heats and cools many times, it warps, and some of the solder balls will crack and stop conducting.
One fix is to heat up the GPU with a SMT rework station, and reflow (remelt) the solder balls so that the broken ones will flow back together and fix the connection. The limitation is that the lead-free solder used on the 360 is hard to work with and is brittle (why they broke in the first place).
The better alternative is to actually remove the GPU/CPU from the board, remove all the solder, put fresh solder balls on the chips, and reinstall them on the board with a BGA rework station.