When my girlfriend starts her Neon, it will sometimes tick for a few seconds. It stops after a couple of seconds and doesn’t happen every time but its not a noise it should be making. I’ve bought some engine cleaner stuff that goes in with the oil and I’ll see how that does. We get oil changes on it pretty regularly, usually around every 3 to 6,000 miles.
If it’s at the first start of the day …or takes several hours of shut down to return, it’s most likely your oil filter’s nitrile Anti-Drain-Back-Valve (ADBV). It’s to slow the drain back of the filter so you don’t get a back flushing of the media upon shut down. It’s peculiar on a NEON since the filter is pretty much at sump level. Anyway, if she has the option where she gets her oil changed, ask for a top tier filter with a silicon ADBV. It stays flexible. The cheaper nitrile valve hardens a bit after so many heating:cooling cycles. The noise, if this is the origin, isn’t damaging as much as it is annoying. Most think that the ADBV is supposed to keep the filter full. While that may be a byproduct of having one, it’s not its primary purpose. If there was any merit to any damage, then every engine manufacturer would absolutely prohibit the use of nitrile ADBV’s …which none do.
Now for other ticks and clicks (if this fails), I recommend Auto-Rx (google it). Assured to clean up anything that’s fouled.