Typically, the woman in the White House is the "first lady." But for the first time in the United States, we may be addressing that woman as "Madam President" in 2017. Gender and the words people use to describe women have been prominent themes throughout the campaign season, and not just because Clinton is the first female presidential nominee of a major party. She has faced sexist attacks from detractors, and her opponent's record of gendered insults and aggressive language seems to be forever expanding. In keeping with the gender theme, why is the term "Madam President" instead of Miss, Mrs., or Ms. President?
The answer is in the definitions of all those words. Merriam-Webster gives two basic meanings of the word "madam": "used to politely speak to a woman who you do not know" and "used when you are speaking to a woman who has a high rank or position." There's an element both of formality and of respect for authority.
This stands in stark contrast to the honorifics "Mrs." and "Miss." Both are characterized by a woman's marital status, though "Miss" also carries connotations of youthfulness. If Clinton went by "Mrs. President," there would be an emphasis on her marital status that just doesn't come along with "Mr. President." She could choose to go with "Mrs.," but we ought not assume that a woman wants the fact that she's a wife to be central to her title as president.
Perhaps "Ms." would serve as an acceptable alternative to "Madam" from a feminist angle. Though the word's originator in 1901 only created it to avoid the embarrassing situation of referring to a woman as married or unmarried when the opposite was the case, it was taken up in the 1960s as a way of addressing women without reference to marital status on principle. Some may be perfectly comfortable with the pairing "Ms. President." But it does lack the recognition of rank and position that tinges "Madam."
"Mr." kind of already comes with an air of authority. It's detached from a man's marital status, and it's not a word that men's rights activists had to invent as an alternative to some other historically used honorifics that were so attached. That's because men (with exceptions based on race, class, sexual orientation, and other demographic factors) have long been acknowledged as autonomous beings capable of leadership. Currently, women comprise only one-fifth of Congress and fill fewer than a quarter of seats at the state and local levels.
Maybe a title with a little extra acknowledgment of rank, such as Madam President, is thus fitting for a female head of state. Should Clinton win, the choice will be hers.