"Republican motherhood" is a term from the time of the American Revolution and shortly after. During that time, of course, the US was becoming a republic instead of a colony. Americans came to feel that their children needed to be brought up differently in a republic. These children needed to be brought up in a way that would make them good citizens of a republic (as opposed to subjects of a monarchy). Republican motherhood was the idea that women were supposed to teach their children the skills and values that would make them good citizens.
This idea has been seen as a step on the way to women's rights. Historians say this because this was the first time that women were credited with having a major role in the health of their country. If the country was to thrive, the idea went, the role played by women would be crucial. Women would need to do a good job of raising and educating their children for the sake of the country. This was a major responsibility.
Republican Motherhood was an idea linked to republicanism that elevated the role of women. It gave them the prestigious role as the special keepers of the nation's conscience. Its roots were from the idea that a citizen should be to his country as a mother is to her child.