Hey OP, it's good that you're asking about this now since places fill up fast way ahead of time.
Do a search for "summer cabin rentals in new hampshire" and you'll get lots of ideas. Websites like VRBO have a bunch of listings.
St. Gaudens is worth the visit even thought it, and Cornish, are really not near anything else. Sunapee and New London are pretty nice areas and would give you a decent base of operations. Most people though, head for the White Mountains/Lakes area in the central and north-central part of the state.
Depending on what you want out of your visit, keep in mind that the southeast quadrant of the state is busy, lots of things to do, restaurants, entertainment, traffic. The central part is touristy (and busy as well). Things get much quieter and more peaceful as you head west and north.