These may not be as wilderness-oriented as you might be thinking, but would be fun at this time of year. None of these are particularly close to you. I think the average drive is 3 hours. However, all of them are feasible hikes for right now, given weather and the fact that you are introducing someone to hiking.
Tuckerman's Ravine. Head up to Gorham NH. Car camp and then the next morning head over to Tuck's with a daypack and those tiny lightweight sleds that are just flat plastic seats. You will not be alone, as there will be crowds of backcountry skiers and snowboarders hiking up with you. Reach the ravine, unpack a picnic lunch, watch some crazy skiing and then go sledding! This is really, really fun and memorable. Have dinner at your camp or in Gorham. The hike is mildly strenuous but relatively short. We actually took a ski trail back down and sledded part of it. I love driving up to NH and this is a guaranteed good time.
Bash Bish falls on the MA/NY border is very pretty and while I have not camped there, I am sure that you could figure something out. It's definitely no Yosemite, though!
http://www.mass.gov/dcr/parks/western/bash.htm
The AT in MA – I would suggest Mt. Greylock. I'd also aim for a weekend that has a really great weather forecast. The beauty of AT hiking is that you can stay in shelters if you want, and this time of year they will not be crowded, and could even be empty. I'd get the AT maps for MA and plan accordingly.
Bear Mountain in CT (right near the MA border) is also on the AT – a very pretty hike, moderately challenging, meanders along a stream once you descend. There is campsites rather than a shelter at this location. Again, AT maps will help you plan this. Kent is a lovely nearby town where you could stop and get a meal after. Lots of quaint small New England towns near by, depending on what route you drive.
Stratton Mountain in VT. Vermont is known as Vermud at this time of year so consider yourself forewarned! Stratton Pond is lovely in the fall and there is a firetower with great views at the top of the mountain. Again, this is located on the AT. It's a very good hike in terms of what your needs are but I think the mud will be horrendous.
Monadnock, NH – you'll actually be above treeline here. I'm not sure what the conditions would be, but this would be the most challenging hike. I generally do it as a dayhike and so can't suggest shelters but it is a part of the M-M trail. You could always call the rangers and see how the weather has been/what your options would be for camping. This is actually pretty close to you – a little under 2 hours.
None of the scenery will even come close to competing with the Whites and Maine. But you can save those for the summer and fall – something to look forward to!