Periodic ignition problem

I hope someone can help! I have a 2004 Saturn Ion, standard transmission, that periodically likes to leave me stranded. Every few months, when I turn the key, it will half-way “turn over” and then stop. Sometimes I can immediately turn the key again, and it starts just fine. Other times I can turn it for 10 times, and it either half turns over, or is just a “click”. If I let it sit for an extended period (20 -60 min) & then come back & try, it starts just fine. I have it to 2 garages, but they couldn’t find anything wrong & said I had to bring it in when it was acting up, which I can’t predict. What do I do, I can’t depend on my car not to strand me!

I would clean the battery cables (both ends of each cable) and I would consider the solenoid.