There really isnt a low cost housekeeping lodging option in Gorham that I am aware of. Town and Country in Shelburne has some efficiency rooms with some kitchen equipment but they tend to rent them out for longer term business renters. I talked to a construction worker on the wind farm that got a $150 per week rate for a room at T&C.
Royalty also has some efficiency units, but they to tend to be booked by long term renters and the hotel is the most expensive in town. There is one more motel next to the chinese restaurant (Gorham Motor Inn?) that advertises fridges in some rooms.
The reason many motels close in Gorham in the winter is that there isnt enough demand and the cost of heating rooms that are inevitably poorly insulated is a big cost. The Mt Madison is a special case, it has been for sale for a couple of years and I think the owners are looking for some time off.
There is another small motel in Shelburne NH on RT 2, I expect if you talk to the owners you can work out a deal as they live next door and dont seem to be real busy. Even if you have to buy a microwave and a dorm fridge at walmart, if you can leave it there during the week it may be worth it as if they are paying off loans, it would be steady income.
An off the wall suggestion is the Philbrook Farm Inn, no efficiency units but it may be worth giving them a call and working out a deal as its their slow season and its hard to beat the history of the place. The owners live in the Inn and you cant beat the view.