The Civil War helped the cause of women's suffrage in a couple of different ways. One of these ways was the fact that women had the ability to do the same jobs as men. Since the men were out fighting the war, someone had to step up and take on their jobs, usually the women.
Not only did women take on the male jobs at home, but they also stepped up and helped out on the battlefield. Some of these battlefield jobs included caring for the wounded, providing supplies to the men in the camp, and even some women stepped up and fought on the battlefield.
By showing the men that they were capable of doing the same exact things they were, it backed up the equality aspect in society.