Mount Mitchell is the highest elevation point east of the Mississippi at 6,684. Which puts Burnsville & Spruce Pine near it for towns. Not a lot to do especially for kids. A lot of the tourist stuff shuts down so the mining places won't be open. The problem is no where is reliable for cold weather and snow in NC. I have hike the Mitchell trial that climbs 3600' in 5.5 miles, which meant I started out in t-shirt and light outer shell on nice clear vegetation no hints of snow or ice to finish up at the top with a foot or so of snow.
I used to Ice Climb, and I would have to call a friend up there to get the "frozen outside dog's water bowl indicator" for if there was a chance the water was turning into climbable ice. If the bowl frozen over night and stayed frozen during the day, there was good odds the ice would form. Multiple days of that happening and/or staying was drive up now or miss it. When making plans to go ice climbing , I often had to switch to regular rock climbing. All the climate protect I needed was T-shirt because the weather was so warm. NC can be frustrating for the weather. Which can be in 60-70's one week and in the 20-30's the next at higher elevations. If the weather is cold enough, the ski slopes might be able to create snow but even that is a gamble if planning. For better odds of snow you have to header farther North. The state is broken up from West to East as Mountains, Piedmont(foothills), Flat lands and then low lands and then the beach and outer banks.
Also be aware if there is any ice they shut down sections of the Blue Ridge Parkway. You can ride bicycles on the sections closed. Contacting the Forestry Service rangers would give you good current conditions.