For any family camping in
New Hampshire, the "Lost River Valley campground" is the perfect choice. It is located in the Western White Mountains Area, less than 4 miles West of I-93 and the village of North Woodstock. This wonderful camp site is very close to the attractions which include Lost River Gorge, Loon Mountain, The Flume and the Clark’s Trading Post. Golfing and canoing can be enjoyed in nearby areas. If you want to reserve in advance then call at 1 800 370-5678, reservations are accepted for over 3 nights stays with 3 night deposit amount. It's such a splendid spot, so don't miss it.
My other suggestion is the Lafayette Campground, White Mountains, New Hampshire. It's a pretty location too and it has a separate bike path which is only open for pedestrians and bicycles, so it would be better for your kids safety. Also a lake like Profile Lake near this Campground to enjoy but i'm not sure to swim. But there are many outdoors recreation available nearby such as fishing, whitewater paddling, hiking and swimming, so probably your kids won't get bored. Hope this helps and have a nice Camping trip in White Mountains.