At any of the White Mountain attractions, you should check ahead of time to make sure they are open on a weekday. Lots close down during the week after Labor Day because the kids go back to school. And the tourists go home!
Alpine Slides are definitely closed. My DD works there in the summer and their last day was the Monday of Columbus Day.
The Conway Scenic Railway is fun. They have trains during the week until Columbus Day weekend. Their web site is
www.conwayscenic.com if you want to check out the schedule. Once in the village area of North Conway for the train ride, there are a lot of shops. And a pretty nice playground for the younger kids.
Shopping always fills up time. Although not a ton of fun for young children.
Instead of driving to the Cannon Tramway, you could check on the Wildcat Gondola's. They go to the top of the mountain and the views are quite spectacular. And it would be much less driving. While going up Route 16 to get there, you'll go by Glen Ellis falls which is a nice walk and really pretty that time of year. They also have a zipline at Wildcat that the 12 year old would probably love!
If you want to do the Mt Washington Auto Road, it really isn't that bad on your car. I've done it a few times with no problem. Bring warm clothing though it will be cold at the top of the mountain that time of year.
If you are staying at Attitash Mountain Village, they have a pizza place called Matty B's that is just kind of OK. The Spaghetti Shed has better pizza but I think she is closed on Fridays now. Will be open for the holiday weekend though.
The apple cider donuts at White Mountain Cider are good. If you take the Conway Scenic, one of the trains has a regular stop there in the fall.
The road someone mentioned earlier that goes from Bartlett to the Kancamagus is the Bear Notch Road. From where you are staying, just turn left and go to the blinking light (only one in town except for the traffic light at Glen Corner). Turn left onto Bear Notch. There is a vacant lot where the Bear Notch Deli burned down on the corner. There are a couple of scenic overlooks on the road that have nice views.
Hope for good weather! We haven't had a great tourist season this year and lots of people are hoping for a big Columbus Day weekend.