Barbara Schierenbeck, a 59-year-old nurse in Brooklyn, is swept up in the excitement of potentially electing Hillary Clinton the first female president. She cannot understand why her 19-year-old daughter, Anna, does not feel the same way.

“Fifteen or 20 years ago, no one would even think about a woman being president,” Mrs. Schierenbeck said. “Certainly, when I was 20 years old in the 1970s, I don’t think I would even have thought about it.”

But for her daughter, electing a woman, while a nice idea, is not a motivating factor. “I want to see someone who, like, has the fervor to fight for me,” Anna Schierenbeck said. A woman will be elected president “pretty soon” anyway, she said, regardless of what happens in 2016. Why does that woman have to be Mrs. Clinton?

The mother-daughter debate unfolding in the Schierenbeck household reflects a debate taking place across the country, as women of varying ages and backgrounds confront the potential milestone implicit in Mrs. Clinton’s bid very differently. As her chances of becoming the first woman to be nominated by a major political party improve, many women are considering how much gender should play into their decisions to embrace Mrs. Clinton’s candidacy — or not.