Consider the map in the section Turn of the Tide on Wikipedia's Women's Suffrage in the United States article: https://en.wikipedia.org/wiki/Women%27s_suffrage_in_the_Unit...

It's not a coincidence that as you go from west to east, the level of suffrage for women generally decreases prior to the 19th amendment. Nor is it a coincidence that the 19th passed. This was due, in part, to the federal system of government allowing states great freedom in how they operated, but also due in part to the expanding frontier. Governments were allowed to experiment. Wyoming (followed quickly by Utah) was the first state to recognize women's right to vote. It actually working was great evidence for proponents in other places. It also put competitive pressure on other states to get it right if they wanted women to immigrate to their state (which they did).

Over the short term, we'll probably see something on the moon more like the early British colonies. But over the long term, once the moon becomes more or less self sufficient and people start spreading out across the surface to found new domes or tunnels or whatever form colonies eventually take, I suspect we'll reap social and governmental benefits.