The new forest is a protected national park. Many of the campsites in the heart of the forest have no facilities. If you have a porta loo then you are allowed to camp. No loo, no entry, take note. These sites are often secluded and very quiet and you will see lots of wildlife.
Sandy balls is on the edge of the forest and in a less popular side of the A31, but none the less still pretty.
Hollands wood is a nice site. Walking distance to pubs and restaurants and Brickenhurst village but at the same time in the heart of the forest.
The Pig restaurant in Brockenhurst is fantastic if you fancy a treat and is walking distance from site (maybe not in dark as you have to go across a field).