Dodge - Neon :: 2005 - Engine Won't Turn Over Sometimes

My stepson's 2005 Dodge Neon sometimes won't start. Turn the key, the dash lights and radio and electric stuff respond, but other than a click there is nothing of an engine start up going on. He'll turn the key a few times, he pumps the gas pedal, does what he thinks might wake it up. Usually it will start after a few tries. He thinks it gets worse in the cold weather, but not much worse.

He also notes that if he does a lot of short trips, like running errands, this problem comes up more often. The car may run fine for weeks, and then the problem hits. Of course, it never does this when the mechanic has it. The mechanic has no idea what to do, since the car works when they test it. I was wondering if I used a voltmeter could I check some key nodes when the problem comes up. Also, if it really does get worse at cold, can I spray some suspicious locations with freeze spray to try and instigate the problem?