We have a saying in Northern New Hampshire and Vermont, "If you don't like the weather, wait a minute." This can wreak havoc with the best laid plans for a vacation up here. My advice: make loose plans, and let the weather decide what you do, and when. Many activities are obviously entirely dependent on a lack of precipitation to be enjoyable. Others, being mostly or entirely indoors, can be enjoyed in a gentle rain or a downpour. Santa's Village and Clark's Trading Post have covered most of their rides and displays, so as long as the rain is gentle with little wind, you can move among the different areas with an umbrella. Actually I prefer to visit these two parks on days with gentle rain as the lines are shorter!
You can take a ride on the Hobo Railroad in the rain, the cars are totally enclosed.
Heritage New Hampshire is entirely indoors, as are all of the museums listed here unless otherwise noted. Rather than letting the weather spoil your vacation, let it make it more varied.
If you have kids, a good way to get the most from your valuable time is to alternate days between visiting theme parks and museums. The kids get a little education, and you don't get burned out!