I think we were all geniuses when we were 17, or at least we thought we were.
Keeping the nephew from cutting wires inside the machine would be a good first step. Maybe he was trying to substitute one of the other counters for the one that is intermittent. Instruct him that before making any changes to wiring he should record the way the wires are to begin with, either on paper or by taking a few photos. Everyone has a digital camera nowadays, it would have been easy for him to record how the machine was wired before he began. Having that information would allow someone to easily put the wires back the way they were.
A lot of people tire of hearing the counters clicking and they look for ways to quiet them, like disconnecting them. You can sometimes do that on older machines but on many newer machines if you disconnect the counters the machine will detect this and give an error. This is because the counters record essential financial information for the casinos and the state gaming regulators. Afterall, the state collects a portion of the money each machine makes. If someone (casino employee) could easily disconnect the counters they could take money from the machine and the casino and state wouldn't know. Most newer machines have "soft" counters within the machine software/ram that records the same financial information, in case the hard meters fail or someone disconnects them.
On the counter that is flaky and not always advancing to the next count there isn't much you can do to repair it. They are usually sealed units to prevent tampering, that's another security requirement. So the best thing to do would be to replace it. There are many of these machines out there so finding a replacement counter shouldn't be difficult.
Maybe someone will have a similar machine or be familiar with yours and can help with correcting the wiring for the counters. You may need to post clear photos showing the backside of your counters and the wires. You didn't specify what machine you have, is it an IGT "S" or "S+"?