I finally found some time yesterday to take a closer look at the furnace.
Gas Pressure
I started by measuring the incoming gas pressure using a gas pressure gauge. I found it to be 6 WC (inch water column), which is well within the acceptable range specified on the furnace.
Dirty Gap
Next I took a closer look at the spark gap itself, and watched closely as the spark was generated. I even recorded the spark with my video camera, which allowed me to zoom in fairly close. This is when I noticed a bit of carbon buildup on the electrodes, and wondered if it could be that simple...
Cleaning the Gap
I grabbed a bit of 220 grit sandpaper, and cleaned the electrodes up a bit. steel wool is probably the tool of choice for this job, but I didn't happen to have any on hand at the moment (not yet unpacked).
Success!
After testing the pilot ignition a few times, the problem seems to be fixed. We had a cold night last night, and the furnace fired up fine all night. As I write this, the furnace has just cycled successfully yet again.
Update
I found this YouTube video from Grayfurnaceman that describes the problem exactly, and says that the only solution is to replace the assembly. I'll be replacing the pilot assembly once the weather warms up, so we'll see if that solves the problem.
Best Answer
Since this question was asked 1 year ago, I have to assume you have solved the problem. If you have not I will give some ideas. first the 360 millivolts you show should have been 750 to 800 millivolts since this is a pilot generator. You can test the rest of the system by temporarily replacing the generator with a 1.5 volt standard flashlight battery. The condition of the wiring is also suspect; loose connections, old oxidized control contacts, an old thermostat, are all potential problems. I would recommend replacing all the wiring and controls and the pilot generator. As for the gas valve it is probably a "general controls" valve but any 800 millivolt gas valve can be used as a replacement. I recommend you call a heating expert. For the battery test above only connect the battery for a second or two just to see if the gas valve works.