I like Moose Brook Campground and have used it as a base for the Gorham area. swimming, showers, field for tossing a frisbee. No hikes out of the CG but Gorham as a base is unquestionably the ultimate as it offers unlimited choices for hiking.
For isolation (not the mtn.), I like the Wild River Campground. Only 8 sites but about as remote a CG you can get to with an auto in the northeast. No extras, just tables, water pump and outhouse. And of course the Wild River for entertainment. Great base for alot of hiking right out of the CG. The 15 mile Moriah loop from the east right out of the CG is one of the hikes nobody I hear talks about, but is full of fantastic views and I love wading the Wild River at the end of the hike after coming off Shelburne Moriah. At the start, Moriah Gorge is really worth exploring especially at that time of year (mucho water). You can reserve a site in advance at the CG there. This is a National Forest Campground.
Last year I stayed Memorial Day week at Dry River and enjoyed my stay there. Frankenstein Cliff is an interesting circular hike right out of the CG and complete the loop with a descent to Arethusa Falls and down the Bemis Brook Trail. This like Moose Brook & Lafayette are state park CGs. The sites here are very big and private especially those further away from the entrance.
I was going to stay at Campton this summer when I planned on finishing my Whites (all peaks southof the Kan), but when checking it out, my group required showers and two showers for a whole CG (nevermind 8 of us) just don't cut it. So I moved my designated trip CG to the private Pemi River CG, sight unseen. I know they have 5 showers and I read somewhere on the web, they have unlevel sites, but I couldn't tell if that is a factor because they were motor home campers vs pup tent campers. So I take a chance. It is convenient for Waterville Valley & Moosilauke hikes and right off the interstate. I hope the river sites drown out any traffic noise.
I also stayed at Cold River CG south of Evans Notch, which was nice and another out of the way WMNF CG with no frills.
I found Franstead CG in Franconia to be crowded. Small tentsites and not much privaccy. But enjoyed the swimming in the river there.
Big Rock off the Kancamagus was OK but excellent location for hikes to the Hancocks, Flume, Osceolas & Owls Head.