Intermittent starts, distributor?

Hello,

I'm a diesel mechanic (and a poor one at that!) So I have some questions for those that might know a little about spark.

The car: 1986 Toyota corrola.

The Problem: Starts sometimes but not other times. This all started 3 days ago. My girlfriend went to work and the car would not start after work. She jumped it (I'm very proud, though she did complete the circuit by grounding at the battery) and still no start. She called me, by the time I got there it was running.

What I have done so far: Battery was old so I replaced it.

When I try and start, sometimes everything is fine. But sometimes, I turn the key to accessory and I hear a real weird nosie under the hood, low volume. Then it will not start. When I hit the starter nothing happens, not even that dead battery ticking niose from the starter.

I am assuming that an electric part is failing but not completely failied. I cannot think of a way that the starter switch would do this. So I started to think about the distributor. Since it moves, I could see it fail;ing selectively. But I don't know anything about gassers. So, what would you guys start testing/replacing and how would you test it?

Also, at the moment I am having difficulty reproducing the problem but I know it will happen again, soon as she is far from home on a cold night.

Thanks,
Steve