--There are a ton hostels (usually around $20-$25) in the Mt Washington Valley area, but surprisingly not in the Franconia Notch area. I know there's one in the Plymouth area, although that's around 45 minutes south of Franconia.
--There's the AMC Highland Center on Rt-302 between Bretton Woods and Attitash, which would put you closer to places in the Mt Washington Valley area and Maine, but would add around 35-40 minutes to any places in VT. I was there recently, and I think it was around $40 for a bunk & breakfast for non-members.
--The Lonesome Lake AMC Hut in Franconia (around a 800' vertical hike part-way up Cannon/Kinsman) is in self-service mode during the Winter for I think around $30 for non-members. However, I've never been there in the Winter, and don't know if that would be too dangerous if you're not with a group.
--I don't think you're going to find anything cheaper than finding 2-3 people to go with, and splitting the cost of a cheap motel/hotel (a few on Rt-18 between downtown Franconia and Cannon; a bunch on Rt-3 between Franconia and Twin Mountain; several in Littleton). Vacancy fills-up quickly in this area during winter, though, once the snow hits--not just from skiiers, but from the snowmobile crowd.
While not a ski town itself, Littleton is actually one of the best towns to "setup shop" for skiing northern VT, NH, and ME, because it's at the crossroads of i-93, rt-302, rt-2, and next to the i-91 junction in St J. I never head north with the idea of going to any one resort--I just go to Littleton and setup shop (I grew-up there, so I stay with relatives), and then decide each day which place I want to go to based on weather and snow conditions (and who is running two-fer deals that day!):
--Four places within 30 minutes: Cannon, Bretton Woods, Loon, Burke
--Six within 45 minutes: above four, plus WV Valley and Attitash.
--Ten within 60 minutes: above six, plus Tenney, Cranmore, Black, and Wildcat
--75 minutes not only gets you to all the bigger ones in northern VT (Jay, Stowe/Smuggs, Sugarbush/MRG [maybe a few minutes more for these two]), it also gets you the other way to Sunday River, as well as up to Orford in Quebec, or at least it did before the border crossing became an ordeal.