You can (probably) do much better than Mount Washington, both in terms of latitude and air quality. For one, you're right about the weather. According to Wikipedia, it is a convergence point for several storm tracks and next to a breeding ground for low-pressure areas. This not only means a lot of clouds, but very turbulent air, which also lowers the quality of the sky. Also, at just a little shy of $2~km$ in height, you're going to find yourself in the middle of all these clouds, which won't be pleasant. Oh, and you should also consider the difficulties of actually building an observatory in such weather.
That being said, building a telescope at very high latitudes is certainly not a bad idea. On the other side of the globe, there is Ridge A on Antarctica, which is agreed to be "the best suited location on the surface of Earth for astronomical research".
Yes, being located on low altitudes allows you to see more of the sky, but you can never see all of the sky at once. Observatories near the equator are great for studying a vast number of different objects over the course of the year. If you're doing a more specific study though (for example, if you want to observe a certain object every night during an extended period of time), you'll have to find yourself a more northern (or southern) observatory. On these higher latitudes, there is a bigger portion of the night sky that is visible throughout the year (constellations in these areas are called 'circumpolar'). This may be desirable for some specific projects.
Looking for good northern spots to build a telescope, we can indeed start by looking at mountains in the northern part of the world. If you take any topographic map of the northern hemisphere, you should immediately notice Greenland sticking out. The big issue with Greenland is that it's covered in a giant layer of ice. You can't build on ice; you'll get in trouble when it melts. We'll disregard that for now though, and we're interested in its highest peak: Gunnbjorn Fjeld. We're in luck here; Gunnbjorn Fjeld is not covered in ice, but is a rocky peak protruding through the ice. The major difficulty would be actually building there, but I'm going to take a wild guess here and claim that modern building techniques and machines could do it. So this is probably what you're looking for. I'm not an expert on the weather in Greenland though, so I can't say anything about that (I do know that it's cold, which is good, because that means the air is stable).
Alternative locations at northern latitudes would be Iceland (but that's probably a bad idea given its geologic activity) or Siberia. The highest point of Siberia is Klyuchevskaya Sopka (which is also a bad idea given that it's an active volcano). Apart from that, googling mountains in Siberia yields lots of peaks which may or may not be well-suited for an observatory. You could probably find a mountain in the Urals as well, but they are not as high as one would ideally want.