OK...
Here's my take. Of course the electrical stuff (ign switch, battery, connections, wires, and all that should be verified as functioning. But.. I'm willing to bet starter. Here's why. As the starter is used over its lifetime and the brushes wear, several things happen. Depending on the design of the starter and it's orientation, the carbon shed by the brushes as they wear builds up in the vicinity of the brush holder (no relation to Eric) and even sometimes, if there is an oil leakage issue as from a rear seal, collects oil.
Either way you end up with a mass of carbon or carbon-oil gook that eventually causes the brush to lose contact with the commutator. When this happens no current will flow.
Since in most cases, the 'ground' side of the solenoid coil is actually the 'hot' brush on the starter (trust me on this), the operation of the solenoid is also dependent on the brushes making contact. One of the signs of pending starter failure is intermittent starting, typically but not always heat related.
If you are spry enough and the starter is in a location that you can try this, carry a hammer - not a 10 pound sledge, just something massive enough to jar things you hit with it (even a big screwdriver handle will do). If and when the beast decides to balk, take the flogging device and bang on the frame of the starter. If it is the brush scenario, this will typically jar the brushes enough to allow the starter to operate (and it will work for quite a few times giving you a chance to arrange for a replacement.)
Of course, as the condition continues, instances of failure will begin to occur more frequently until even flogging it with that 10 pound sledge won't work..
But - if it quits and you bang it with the flogging device and it starts, you're definitely looking at replacing the starter..
Wellnow.
My first impulse was “starter,” without really knowing why or how I came to this “hunch.”
Because of the fact that the intermittent starting takes place in the heat (usually a warm day, with 25 minutes of driving the speed limit on the freeway) and I know that some race cars will not start after a run because of the proximity of the headers to the starter, Busdaddy and I both thought the starter is the culprit.
I have seen people hit the starter (not with a 10# sledge!) and were able to then start the vehicle. However, I would prefer not to resort to this method as it is only temporary.
Not having a manual on hand for the engine/etc specs, I’m not sure where the starter is located, but I’m sure if I went out there with some determination and a few prayers, I could find it.
Whatever the problem is, it is becoming more frequent, which again, as you suggested, could mean the starter. So I will price the little wuzzers and try to get the vehicle up and running again without too much angst.
In cooler weather, there seems to be no problem.