Green was born in Brampton, Derbyshire, England. Her mother had joined the LDS Church three years before Green's birth, but her father never joined the Church. She emigrated to the United States with her mother and some of her siblings in 1889. Green was baptized in The Church of Jesus Christ of Latter-day Saints in 1891, by then living in Salt Lake City. Green was raised in the Salt Lake 5th Ward of the LDS Church. Early on she was a teacher in both the Sunday School and YWMIA. She served as a Mormon missionary in the Central States Mission of the LDS Church from 1907 to 1909. After studying booking and accounting Green began work as business manager for a Salt Lake medical clinic. In 1920 Green was made president of the YLMIA for the Granite Stake in Salt Lake City. She served in this position for the next 12 years, and oversaw the initial establishment of the Gleaner program. In 1932, at the age of 50, Green married Bryant S. Hinckley, whose wife Ada Bitner had died in 1930. At the time, five of Hinckley's 13 children were still living at home. Green was at that point president of the stakeYoung Women Mutual Improvement Association. Gordon B. Hinckley, one of the children, later recalled that he and the other children were upset by their father's decision to remarry, but they eventually came to accept their stepmother: "I don't know that it was easy for her to step into our family, but she did it well. We all respected her. We all loved her". In 1935, when Bryant Hinckley became the president of the Northern StatesMission based in Chicago, May Hinckley went with him and presided over the Primary Association, YWMIA, and Relief Society within the mission. In 1940, May Hinckley was asked by church president Heber J. Grant to succeed May Anderson and become the third general president of the Primary Association of the church. In her -year tenure, Hinckley introduced a revised curriculum, added a scripture-reading program for leaders and teachers, established a formal scriptural theme for Primary, and selected the official Primary logo, motto and colors. Hinckley formed a committee that created lessons for use by Primaries in missions. With energy rationing as a result of World War II, she oversaw the creation of more home-based Primary programs. Hinckley was the editor of The Children's Friend while she was Primary president. Her term ended when she unexpectedly died of pneumonia in Salt Lake City, Utah, the day after her 62nd birthday. She was succeeded by Adele C. Howells, her first counselor.
