So I put a headlight wiring kit from LMC Truck on my '96 Cherokee that circumvents the normal wiring and runs the headlights directly off the battery by way of a relay. The kit runs the lights at 55/100 (low, hi) watts instead of the original 55/60. One reason I did this was because the light output was terrible, and the other was because for some unknown reason the headlights would shut off after being on hi for a bit. Never did this on low. I've had the new wiring kit on for a few years now and it seems to have solved the issue.

The problem I'm having now though is that this new harness is melting at one of the connections...




You can see only one of the wires is overheating while the other seems to be ok. I'm guessing the hot one is used by the high beams.

What could be causing this? Could it be related to why the high beams used to shut off before I replaced the wiring harness? A while back I asked about the shutting off issue on a number of Jeep boards but nobody had heard of this particular problem before.

 

Usually when there is resistance at a connection heat gets generated..
Can you manage to have some one run a current check as to how much power your hi beams might be pulling ?
100 watt should pull approx 9 amps. Two of them about 20 amps. 14 gauge wire is not sufficient for that kind of load. The wire it appears that is supplying the plug is 10 gauge which should handle approx 30 amps. If you want to keep the hi power hi beam lamps you will need to use 10 gauge wire to the relay and 10 gauge wire coming out of the relay to the hi beam connection at the headlamps. From what I believe I see in your pix , the wire is too small and the connections can' t handle the load.
You can either use lower power requirement hi beams or do some special wiring to handle the hi current those lamps consume. Not a big deal, just use 2 relays one for each hi beam lamp . The fused (30 Amp) 10 gauge wire can supply the power to both relays and then when the hi beams are desired it can energize the relays which can now pass the current on to the individual headlamp.
Something to consider, if you used 2 fused power supply wires from the battery positive to each of the relays , you should almost never lose both hi beams at the same time.

Most pull out headlamp switches have an internal circuit breaker that when current exceeds a certain amount, like hi power hi beams, the heat causes the breaker to open and when it cools some the breaker closes and the circuit is complete again.and the lights work as usual.
You could also use automatic resetting circuit breakers instead of fuses to the relay supply and if there ever is any problem you won;t be playing with changing fuses.