Lucky you, I lived in Ringwood (well a small village just between Ringwood and Christchurch) and I loved it - so did everyone else who used to come and stay - all the time!!! I was there a long time, working in Lynhurst, until I moved upt'north because of DH. I would go back in a heartbeat.
Lymington is lovely, but gets extremely busy during the summer and it is a long way down through the forest to get to it, it you get stuck in the holiday traffic, it is a nightmare!
Brockenhurst is very, very small, not much there and again right in the forest, but it is beautiful. New Milton, is slightly easier to get to, and may be a bit cheaper, but further down.
Ringwood is very accessible and has good access everywhere, there is a good little town, yet the beach is just a short drive. It doesn't get too touristy.
Don't forget Fordingbridge, north of the forest. Again, could be slightly cheaper, good access to the forest and to Salisbury, which is a brilliant town.
For my money I would go Ringwood area.