The type of camping you do will dictate a lot of your instrument choices. A backpacker will need an instrument that is incredibly light, while a car camper is not concerned about the weight. If you are camping in the Pacific Northwest of the U.S., you need an instrument that will not be affected by extreme humidity.
Probably the ideal instrument for camping is the harmonica. They are pocket-sized and weigh just a few ounces. Harmonicas are durable enough to survive a dunk in the river or a tumble down a mountainside. Harmonicas don't need to be tuned before you start your evening performance. Quality harmonicas are available tuned in different keys to produce different types of sounds. The harmonica player can't sing while playing, so a partner will need to perform the vocals.
Another popular instrument for camping is the acoustic guitar. Guitars are lightweight, though they tend to be fairly bulky. Several guitar manufacturers offer smaller models aimed specifically at campers, although they may still be too big for ultralight backpackers. It's very easy to find sheet music for the guitar, even for songs that were not originally written for the instrument.
While the harmonica and guitar are the two most common instruments around a campfire, almost any small acoustic instrument would be a welcome addition to an evening sing-along. The mouth harp, or jaw harp, is a good pocket-sized instrument to sound out simple rhythms. Small drums, like bongos, are great for some types of music.
If you do bring musical instruments on your next camping trip, you should be respectful of your neighbors. In established campgrounds, follow any rules for quiet hours. In the back country, you should be aware that sound travels far in the quiet night air. When you have a sing-along around the campfire, you are sure to meet new people as they stop by to join in.