I was in the same position as your son, and unfortunately I think the best solution would be to wait until he has a somewhat related problem that does require a repair shop to do work on the car, and simply ask them to fix the reverse lights while the other work is beind done.

My backup lights went out probably two years ago. I asked about getting someone to fix the lights, but my parents nixed the idea since it would be $100-$150. However, when my tail lights went on the fritz just a few weeks ago, I realized I would have to take the car in to get the brake lights working again. Even though my parents had pretty much told me before not to worry or do anything about the backup lights, I went ahead and told the mechanic to fix them while he was working on the rest of the tail light cluster.

My logic was that the car would be in for *at least* two hours ($65/hour), if not longer (since there was a brake failure light issue as well as a tachometer thing to check on), and that we'd end up spending more than $120 or so as it were. Why not get everything fixed for just a small additional fee?

They fixed it, and my final bill was $180. I doubt it would have been much less if I had brought in the car for only the backup lights--they were already doing work, so just throw some extra repairs in as a package deal. I think the problem was a simple connection issue, but I don't have the technical skill or tools here at college to take apart my shifter (and what if I screw it up?). It was much simpler and not much more expensive to get the work done while the car was already due for repair work.
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'89 244 GL -- 106,520 miles (see profile for info on car)