For what reasons did women campaigning for the franchise in the Nineteenth century want the vote?

You said it yourself: Equality. Men, including black men, could vote, yet women were treated like village idiots, patted on the head, and told to go home and cook supper and take care of the kids--like that's as high as their intelligence level got!!! So we were demanding the equal right to vote. And someday, the Good Lord willing, we'll live to see a woman as President.