Holland’s Wood is good as Legoman says but right by the A337. Also gets very busy at peak times but a short walk from Brockenhurst shops and pubs and on the cycle network.
The Red Shoot pub at Linford has camping at the back. Quiet, nice views, pub on doorstep. Easy riding out the back to other pubs. About 10 mins drive to Ringwood for shops and 20 mins to Southbourne beach.
Ashurst has 2 pubs right on the door, a good Indian, Chinese takeaway and chip shop. Cycling from the campsite. A bit further to the beaches, about 40 mins. Lots of great cheeky trails I can show you (I live there!)
There’s a new seasonal campsite at Lepe beach. Right on the beach, not sure of facilities. New cafe, just up the road to the forest, 10 mins from beaulieu and pubs.
Hope that helps!