Why would this A-10 Thunderbolt be flown over rural New Hampshire?
Practice makes perfect!

Groton, NH (along with a large chunk of rural New Hampshire further off to the north) sits more or less under an area of military Special Use Airspace called the Yankee 1 and 2 Military Operating Area complex, or MOA for short:



(If you're wondering why Groton is not on the map, ask FAA/NACO -- it's about at the bend in the road WNW of Hebron, right under the southern edge of the Yankee 1 MOA.  The snippet was taken using vfrmap.org.)

These areas of airspace are used by US military tactical pilots to practice their tactical maneuvers (such as "dry" practice bomb and strafing runs, and air combat maneuvering) without getting in the way of other airspace users -- high-performance jets making sharp maneuvers don't mix well with slow little propeller planes or not-so-maneuverable airliners!

As to why the pilot was circling?  They were probably killing time, waiting for their practice partner to show up, or the MOA to free up so they could use it.  Also, the airplane in the picture looks to be unarmed -- on one side is an empty pair of Sidewinder rails, while the other side carries a pair of targeting and countermeasures pods.

So, what your mom saw was a small part of our military keeping their gun oiled and ready, so to speak -- practice makes perfect, after all, and the airspace over rural parts of the US is about the only place in this country non-busy enough to serve as a suitable practice area.