You have a tough issue there for sure and there's certainly enough expertise here to get you through it.
Since you're in the middle of a potential fix, I just want to drop these 3 comments until we hear how your new wire run works out.
1. I am with Dave, due to personal experience, that you really should replace that Dimmer switch. They get faulty, cause grave issues, and they are cheap and easy to replace. Do it.
2. Bear in mind that you very well could have more than one issue. Don't let yourself get stuck on a single solution that solves everything.
3. Personally, I really really think you should find out where that mysterious 2nd small wire is going to from your battery post. It might be as simple as a home-run for a radio, or old switch someone installed, or hell, the key itself. People do weird things. If you can crimp or solder a wire (any monkey can!) than you might even consider just cutting it or removing it at the battery and you'll soon figure out what it was for if it serves any importance.
Cheers! Good Luck! Keep us up-to-date!
Because you all asked for update:
I bought a custom 2 ga battery cable from:
I had them add two 12 ga wires; one to replace the wire to the horn relay; and one if I wanted to retain the 'extra' wire that powered the power seats a previous owner added to the car. It was made up the next day and shipped. Wow, great response time. It arrived in two days; impressed with the workmanship. I installed it w/o connecting the wire to the power seats and the car started immediately and ran perfectly. Everything expect the power seats worked.
Yesterday I removed the driver's side kick panel and pulled up the carpet to get a look at the wiring to the convertible top, the power seats, and the dimmer switch.
The wiring for the top looked original and in very good shape. The circuit ( as far as I followed it) is in accordance with the wiring diagram from the '61 shop manual. There is a 40 amp (marked on the breaker) circuit breaker in that circuit. I intend to leave that as is. It always worked, is working, and I'm NOT going to fuss with it.
The wiring for the power seats was not run though the circuit breaker. It was 'added' (not surprisingly) and connected directly to the seat control circuit.
The dimmer switch look original. The wiring looks original and is run as shown in the documentation. I am going to replace the dimmer. I also decide to replace the 'guts' of the headlight switch as a precautionary measure. I bought a dimmer, the connectors to the dimmer, and the headlight switch from:
Part numbers 500042, 500286, 01995096.
As of today, I do not intend to re-wire the power seat. I bought a circuit breaker (American auto wire part 04850166) and will install that in the existing 'extra' 12 ga power line.
I'm waiting on the parts. When installed if everything works, I'm going to declare success (at least for now).