New Oxford American Dictionary says:
ORIGIN late 17th cent. (in the sense [rudiments, essential elements of something]): from Latin, plural of stamen in the sense ‘threads spun by the Fates.’
1676, "rudiments or original elements of something," from L. stamina "threads," pl. of stamen (gen. staminis) "thread, warp" (see stamen). Sense of "power to resist or recover, strength, endurance" first recorded 1726 (originally pl.), from earlier meaning "congenital vital capacities of a person or animal," also in part from L. application to the threads spun by the Fates to determine the length and course of one's life, and partly from a fig. use of L. stamen "the warp (of cloth)" on the notion of the warp as the "foundation" of a fabric.