The problem you describe is indicative of an intermittent VATS (Vehicle Anti Theft System) issue. It's very common.
A good way to fix this issue is to first get a hold of an 86 GM Shop Manual and a digital multi meter. Once you have that, you'll find there are excellent troubleshooting charts just for this very problem. The next thing you will need is to be in a position to do the checks in the shop manual while the car is in it's little no start condition fit. This will prove to be the hardest part of all, since the problem happens completely at random and can catch you off guard. Otherwise, everything will check out fine when it's starting regularly and you won't find the problem. I had the exact same issue and for me it was the starter enable relay located behind the service engine soon light panel.
If this is your daily driver, I would not put this off. I kept putting it off, and finally the vette never started back up again and left me stranded in a parking lot. This was actually to my benefit. It gave me plenty of time to diagnose the problem.
In case of an emergency, cary a long piece of wire with you in the car. If your vette won't start up after waiting, you can simply jump that wire from the positive battery terminal directly to the small terminal of stater solenoid, after leaving the key in the on position. This will bypass everything and will crank the car so you can at least get home.
A quick way to check it next time it does not start is to do this. You will need a multi-meter and the car must be in it's no starting condition, otherwise everything will check out find and you won't find the problem.
This will tell you for sure if the problem is the starter or something in the VATS.
Next time it does not want to start, unbolt that single wire on the small terminal of the starter solenoid. It's the one closest to the engine. Now attach one end of a volt meter to it, and the other to ground. Have someone in the car try and crank it over. If you don't see battery voltage while the key is held in the cranking position, it's not the starter. It's something in the VATS system. If you do see battery voltage when cranking, quickly connect that wire back on to the starter. Try to start the car again. If it still does not start, it's the starter.