Born in Virginia, Minor became a member of the St. Louis Ladies Union Aid Society during the Civil War. The successful management by women of the complex responsibilities of the society convinced Minor that women deserved political equality. In 1865, when public debate began to focus on voting rights for recently emancipated African-Americans, Minor was the first woman in Missouri to suggest publicly that women should also be given the right to vote. In 1867, she unsuccessfully petitioned the Missouri legislature for that right.

In May 1867, Minor, Rebecca Hazard, Lucretia Hall, Penelope Allen, and Anna Clapp held a meeting at the Mercantile Library and organized the Woman Suffrage Association of Missouri, said to be the first organization devoted to the single goal of women´s suffrage. (Minor would remain president of the WSA until 1871.) At the second meeting, they developed their mission: ´The sole object of this association shall be to secure the ballot for women upon terms of equality of men.´"

The 14th Amendment was passed on July 9, 1868. It granted citizenship to all persons born or naturalized in the United States; and stated that no state shall make or enforce any law to abridge the privileges or immunities of these citizens or deprive any person of life, liberty, or property, without due process of law; or deny them equal protection of the laws. Minor argued that this implicity guaranteed women the right to vote. In October 1872, she tried to register in St. Louis; when she was refused, she brought suit (through her husband, as women could not sue in court) against the registrar, Reese Happersett. The case ultimately went to the US Supreme Court, where the Minors lost. It generated great publicity for women´s suffrage, however, and another amendment to the Constitution was submitted to Congress in 1878. It was not until 1920, 45 years after Minor´s appeal was overturned, that the 19th Amendment finally gave women the right to vote.

Minor remained active in the cause of women´s rights. She was president of the Missouri branch of the National Woman Suffrage Association, and in 1890, president of the Missouri National American Woman Suffrage Association. She died in St. Louis in 1894.