Courtesy of the Online Etymology Dictionary:
"1895, Amer.Eng. (in a Montgomery Ward catalogue), from "Fédora," a popular play by Victorien Sardou (1831-1908) that opened 1882, in which the heroine, a Rus. princess named Fédora Romanoff, was originally performed by Sarah Bernhardt. During the play, Bernhardt, a notorious cross-dresser, wore a center-creased, soft brimmed hat. Women’s-rights activists adopted the fashion. Men began to wear them with city clothes after 1924, led by Britain's Prince Edward (Edward VIII), the most influential man of fashion in his day. The fem. proper name is Rus. fem. of Fedor, from Gk. Theodoros, lit. "gift of god," from theos "god" + doron "gift.""