I've come across heat like that before in computers and it was a bad solder join creating resistance, causes it to heat up enough to cause enough resistance that what ever on the other side wasn't getting enough juice and would cut out. probably cools off enough after to reconnect and continue.

could just be a bad power supply too, might be caused internally in the PSU and you feel it most closest to the plug on it because that's where the heat transfer is going to go (kinda like a heatsink on a computer)

i'd hunt down a N64 power supply replacement and see how that goes, should be able to find tons of them, people don't know what they are after they get separated from the system, then they get donated by them selves. I always pick em up for about a buck or so