It's been quite a few years since I've spent some time in NH, but we go to VT often. Here are a few things on the eastern side, not too far from the NH border.
Montshire Museum in Norwich, VT, not far from White River Junction. Here's the website http://www.montshire.org. We used to take the girls when they were younger. It's a great place for kids ages 5 to 10. Lots to do inside and out. It's a fun, educational experience. Bring a picnic lunch.
Queechee Gorge is worth a visit. If you like to hike, there's a path to walk down to the bottom. It's a steep climb back up.
If you're interested in raptures (hawks, eagles, owls), you'll enjoy the Vermont Institute of Natural Sciences in Queechee, VT.
I've never been, but I've heard the Billings Farm and Museum in excellent.
In VT or NH, stop at a maple sugar farm. There won't be much going on since the syrup is made in the spring, but you can see the equipment and learn about the process. Some farms will give you guided tours.
Vermont state parks have the cleanest campgrounds. The sites are also quite large. The only drawback is no hookups. We've never stayed at a state park with electricity.