mid-15c., donour, "one who gives or bestows, one who makes a grant," from Anglo-French donour, Old French doneur (Modern French donneur), from Latin donatorem (nominative donator) "giver, donor," agent noun from past participle stem of donare "give as a gift," from donum "gift" (from PIE root *do- "to give").
As "person from whom blood is removed for transfusion," by 1875; in reference to those living or dead from whom organs or tissues are removed for transplantation, by 1918 (originally of guinea pigs).