Hi Tom,
I can echo what most have said about NH, with a couple additions...
Hanover CC holds a special place in my heart, as I went to golf camp there at age 15, my wife went to Tuck, and I now count Dartmouth as a work client, so I have played it many times pre and post renovation. The history is neat, but the current course is not noteworthy on anything more than a regional level. The old Barton (possibly) holes across the street now used as practice holes are pretty neat if you know where you are going. There are about 5-6 greens still there, but navigating the multiple corridors and blind hillsides is a bit tough for a first timer...I wouldn't be surprised if a newbie missed one.
Lake Sunapee and Manchester CC are great old Ross courses, and MCC is the perennial #1 course in the state by GD's metrics. The Balsams is a great Ross layout too, and is hidden merely by the fact that it is almost in Canada, and located at the top of a mountain. All are worth a look if time is on your side, probably ranked Balsams, MCC, LS for uniqueness. Bald Peak also gets great reviews (partly because of its Myopia/Chicago Golf-type exclusivity?) but I have yet to play there.
Lake Winnepesaukee, GC of New England, Montcalm, and Baker Hill are the new courses on the block, but I have only played the first two. LW is a treat, but thoroughly modern. Some beautiful wooded landscapes and rock outcroppings throughout, although with blinding white sand... GC of NE is also a good layout from the Palmer group, and wouldn't be out of place in the Western US.
Portsmouth CC is a neat early-mid career RTJ course with several holes along Great Bay, the inland salt water waterway. They hosted the New England Amateur a few years back.
Other courses to research are North Conway CC, Keene CC, Nashua CC, and Wentworth (Rye Beach), probably in that order. None of those will make any "best of" lists outside of NH, but all have some history...NCCC goes back to about 1900, Wentworth is a Ross/Silva, and the other two have Wayne Stiles roots.
Has Attitash mentioned golf again lately?