Randolph and Jefferson are both located ideally but it will be tough to find a house to rent for a week. There are places to rent but usually not advertised anywhere, mostly word of mouth. Randolph is more of a summer colony then a town. It is undeniably in the middle of the northern white mountains and there is a long section of the AT accessible from town plus a very dense local trail network. The trail network is set up for loop hikes or trail runs. Most of the homes and camps are in use by the owners in the summer, I am not aware of any broker who actively handles or markets rentals. The sum total of businesses in town is a gas station with a small store, an older motel with several rooms and a tourist cabin village. If you want prepared food of any type you need to go to Gorham. Jefferson is very similar although set back a bit from the national forest its does not have a trail network, to get a prepared meal except for pizza and sandwiches you need to drive over to Lancaster or drive through Randolph to Gorham.
Gorham does have ATV traffic on Main Street and along a section of state highway but the majority of the town is off limits. They support the same tourist businesses that hikers and other visitors would use. There is a selection of restaurants in town. The biggest issue is ATV folks tend to fill up the hotels/motels especially on weekends. Gorham is also a major thru hiker town as its logical spot to take a break before the Mahoosucs. If you spend any time in town you will see thru hikers going about their resupply. The Top Notch Inn is a strip motel but they have a couple of residential homes that they have rehabbed and rent on a weekly basis.
https://topnotchinn.com/
One spot that few think about that might fit your needs is Philbrook Farm Inn in Shelburne NH.
http://www.philbrookfarminn.com/ The AT runs through Shelburne. Philbrook Farm has family housekeeping cottages (actually old summer homes set off well away from the Inn) for rent. Shelburne has even less services for tourists (not even a gas station) but Gorham is a short drive. There are some extended families that have been meeting up there for decades. A major detraction to some, a bonus to others is I dont know if the cottages have internet service (there is cell service). These are older cottages, no AC.
Lincoln/Woodstock on the other hand have a major ski area nearby and there are hundreds of ski condos in town plus full tourist accommodations. My guess is renting a condo for a week is lot easier. It convenient to Mt Moosilaukee and Franconia Notch but a bit of a drive over to Mt Washington and Crawford Notch. It also has ziplining and boat options (but no ATVs). Its also is the closest to the Pemi which is the center of the whites. The infamous Pemi loop is usually picked up just out of town at Lincoln Woods. I expect your trail runners will have this high on their list. The start of the Presi Traverse which may also be on their list is about 1 hour north in Randolph with the end about a half hour away in Crawford Notch.