The Neon gets really angry if it doesn't have a lot of juice. Checking for corossion is always a first thing to do. High current flow or shorting of a particular device can cause corrosion across the terminals, even if not visibly evident.
If you are hearing a loud "CLICK!" from your starter motor then it means that your starter may or may not be getting enough juice to start your engine. So what you should do is check your starter (and the starter connections)). So to check your starter first disconnect all electrical connections from it. Then remove your battery and it's casing from the car (this requires that you have to remove some screws by accessing from under neath the vehicle, but if you have any tiny friends then you don't even have to jack up the car to remove the starter

). Of course before you can remove the battery casing you have to disconnect the black box with all of those wires with your relays and crap from the side of it, which is fairly simple to do.
Okay, so now you've removed your battery, the battery casing (which includes the holder) and just put that big black relay/fuse box towards the firewall. Now you will be looking at your transaxle. Okay. Your starter motor is mounted with only 2 really long bolts. One bolt will be found facing the drivers side of the vehicle and it goes through the casing of the transaxle. Remove that one. Now the other one is lower by a couple inches and facing the passengers side. It's a tricky one, but you can undo it from the top of the car. Now you can slide the starter out from the left side and out from the top of the car.
You test your starter by applying ground to the case of the starter, a large positive wire to the obvious positive terminal, and then apply positive to that tiny little connection. That's how you test it.
You can pick up a new starter at an auto-wrecker for $69 CDN so probably like $45 USD.
But first clean off all terminal connections really really well and check your starter terminal connections and clean those off anyways too. Just basically clean the shit out of every electrical connector that goes to your starter including your battery (like literally use steel wool and scratch it right up to clean off all bad things). Then if that doesn't work then remove and test your starter.