late 14c., in reference to the grammatical case, from Old French genitif or directly from Latin (casus) genitivus "case expressing possession, source, or origin," from genitivus "of or belonging to birth," from genitus, past participle of gignere "to beget, produce" (from PIE root *gene- "give birth, beget," with derivatives referring to procreation and familial and tribal groups).
This word was misused by Latin grammarians to render Greek genikē (ptōsis) "the general or generic (case)," expressing race or kind (Greek genikos "belonging to the family"), from genos "family, race, birth, descent," from the same PIE root as the Latin word. As the genitive also is the case of the possessor, and the Romans "were not strong in abstract matters" [Gilbert Murray], the result was some confusion.
The Latin genitivus is a mere blunder, for the Greek word genike could never mean genitivus. Genitivus, if it is meant to express the case of origin or birth, would in Greek have been called gennetike, not genike. Nor does the genitive express the relation of son to father. For though we may say, "the son of the father," we may likewise say, "the father of the son." Genike, in Greek, had a much wider, a much more philosophical meaning. It meant casus generalis, the general case, or rather the case which expresses the genus or kind. This is the real power of the genitive. If I say, "a bird of the water," "of the water" defines the genus to which a certain bird belongs; it refers to the genus of water-birds. [Max Müller, "Lectures on the Science of Language," 1861]
The noun meaning "the genitive case in grammar" is from 1610s.