Which term is correct, "woman suffrage" or "women's suffrage"?
"Woman" was meant, in the 18th and 19th centuries, to be a term parallel to the philosophical, political and ethical use of the term "man." Just as "man" is often used to personify and stand for all men in general (and often meant to be inclusive of women as well), so "woman" was used to personify and stand for all women in general.
But there's another subtlety in the difference between the terms. By personifying men or all people as "man" and women as "woman," substituting the singular for the plural, the authors also implied a sense of individuality, of individual rights and responsibilities. Many of those who used these terms were also associated with the philosophical and political defense of individual liberty over traditional authority.
At the same time, use of "woman" implied a common bond or collectivity of all of that sex, just as "man" in "rights of man" managed to imply both individual rights and a collectivity of all men or, if one reads it inclusively, human beings.
Historian Nancy Cott says this of the use of "woman" rather than "women":
"Nineteenth-century women's consistent usage of the singular woman symbolized, in a word, the unity of the female sex. It proposed that all women have one cause, one movement." (in The Grounding of Modern Feminism)
Thus, "woman suffrage" was the term most used in the 19th century by those who worked to achieve the rights of women to vote. "Women's suffrage" was the term used by many of the opponents. In the early 20th century, as the concept of individual rights became more accepted and less radical, the terms became more interchangeable, even by the reformers themselves.