Your question has taken me back to an incredibly pleasant trip to Rome which I had with my family (three people with a slightly older boy than your son) a few years ago in April.
We stayed in an extremely central place which I would recommend to you (Pensione Barrett), from which you can easily walk around Rome, which allows you to stumble into some of the ancient ruins, and which is rather a work of art of itself (both the rooms and the common areas are decorated with statues, for example). The downside of this is the fact that the hotel has a limited number of rooms, so you'd better reserve in advance. Although their site is (I'm afraid) only in Italian, I'm sure they also speak English. The price is within the bracket you have mentioned. P.S. I checked on TripAdvisor, and Pensione Barrett is mentioned as top 22nd address in Rome.
A very good alternative to Pensione Barrett is Residence del Bollo. I think that originally it was part of a religious building, it has now been converted to a set of small apartments, located very conveniently close to Piazza Navona, which would allow you to set your own timing to the walks around Rome. I know that sometimes they cater to groups of students (that's how I got in touch them), so again I suppose it might be useful to book in advance. The prices are normally quite reasonable, although at times they become incredibly steep, so once again you'd better check their site well in advance.
A different option to staying in hotels is to secure a room in a monastery. There are plenty of such institutions which provide accommodation at moderate price in interesting buildings which normally are located in central areas; I gather that they normally have restrictions as for times to access the building in the evening. Unfortunately, I don't remember any such address at present, but will try to come up with one.
Enjoy your visit!