There are several places with campsites near Princeton. The closest is:

Washington Crossing State Park

355 Washington Crossing-Pennington Road
Titusville, NJ 08560-1517
(609) 737-0623

Online reservations can be made 7 days per week/24 hours a day at ReserveAmerica. To make a reservation through the call center, please contact ReserveAmerica at 1-855-607-3075 from 8 a.m. to 8 p.m. Monday through Friday and 9 a.m. to 5 p.m. on Saturday and Sunday from April 1st through October 31st. Beginning November 1st through March 31st, hours are 9 a.m. to 6 p.m. Monday through Friday and closed on Saturdays and Sundays. 

Group campsites: Four group sites with picnic tables, fire rings, and portable toilets; total capacity: 115. Open April 1 through October 31. New Jersey Residents: $3 per person, per night

Non-Residents: $5 per person, per night. We highly recommend that group leaders carefully estimate their group size so not to over estimate as refunds cannot be made later.