While I don't know how far camping is spaced along it, look into the trail through Pine Creek Gorge in PA:http://www.visittiogapa.com/railtrail.html
These folks provide shuttle service over the entire length of the trail, so you could ride one way and have your car waiting for you at the end:http://www.pinecrk.com/
Or you could ride both ways. I have used their shuttle service in the past for backpack trips. There are two state parks with camping (one has primative camping only) near the northern terminus of the trail, which is in Ansonia. As you can see, you could also do some canoeing if you have the time. Nearby Wellsboro is worth a look. Some of the streets are lits with gas mantle lamps.
Another option closer to home is the Missisquoi Valley Rail Trail, which runs between the St. Albans and Richford and passes near Lake Carmi State Park, which is nice:http://www.vtstateparks.com/htm/top5bikecarmi.htm
I stayed at Lake Carmi as part of ACA's Green Mountains Loop. You will have to do a few little hills to get from the trail to Lake Carmi. The trail is only 26 miles, so that might be too short. On the other hand, it could be a good test run for something longer. You could ride from Ritchford to Lake Carmi, then to this place in St. Albans, which looks neat:http://www.vtfpr.org/parks/htm/burton.htm
I think the kid(s) would enjoy the ferry ride.