Could be that the console is not getting sufficient power.
Sometimes the insides of the power switch will corrode over time causing the resistance across the switch to become higher than normal. This would reduce the voltage coming out the other side of it. The power switch for the N64 has legs for 12V and 3.3V power. Check the resistance across both (there will be a silkscreen on the bottom of the board telling you where the traces for the 12V and 3.3V lines are). If it is more than a few ohms, consider desoldering the switch, bending back the metal tabs on the bottom of the switch that hold it together, and cleaning the contacts inside it. This may solve your power issue.
Another possible cause is a bad/corroded solder joint on one of the legs of one of the chips inside the console. When the system runs for long enough, the chips will warm up causing the metal legs to flex slightly with this heat. This may cause a loose leg to break electrical contact with the board. A reflow of the CPU, or RCP (biggest chips with the smallest legs) with plenty of flux, and some new solder may help. If you’re going to attempt this, you should have some no-clean flux, solder wick to clean up bridged pins, and something to take excess solder off your iron, like a brass sponge. Look up techniques for ‘dab soldering fine pitch qfn’, or ‘drag soldering fine pitch qfn’. It is important to be gentle with the pins of these chips, as they are quite easy to bend if using a heavy hand, or an oversized tip on a cheap soldering iron (the included chisel tip on a Weller SP25N would be an example of an oversized tip for this job).