c.1400, "an end, settlement, retribution," from Middle French finance "ending, settlement of a debt" (13c.), noun of action from finer "to end, settle a dispute or debt," from fin (see fine (n.)). Cf. Medieval Latin finis "a payment in settlement, fine or tax."
The notion is of "ending" (by satisfying) something that is due (cf. Greek telos "end;" plural tele "services due, dues exacted by the state, financial means"). The French senses gradually were brought into English: "ransom" (mid-15c.), "taxation" (late 15c.); the sense of "management of money" first recorded in English 1770.