Both in the White Mountain National Forest and run by the AMC:
Joe Dodge Lodge in Pinkham Notch. I've done MANY ride-n-hike trips in the Presis from here, the desk staff are quite friendly and have been willing to store my gear in an area not accessible to the general public while I'm on the trail. The rooms sleep four in bunkbeds, private rooms are available for a bit more $. Dorm style bathrooms and showers. No beer or wine for sale, but they do allow BYOB. Breakfast, lunch and dinner are available on site - the kitchen does a breakfast buffet, lunches to order and family-style dinner. I've yet to have a meal that isn't excellent.
Highland Center at the top of Crawford Notch. Very similar to Joe Dodge, but much newer. They do serve alcohol in the dining room. A bit spendier, but again well worth it.
These places are hiker-oriented, but still quite moto friendly. Be prepared to answer a lot of questions from curious guests, think of it as an opportunity to build some goodwill among the non-moto public.