c.1300, "strap fastened to a bridle," from Old French rene, resne "reins, bridle strap, laces" (Modern French rêne), probably from Vulgar Latin *retina "a bond, check," back-formation from Latin retinere "hold back" (see retain). To give something free rein is originally of horses.
c.1300, from rein (n.). Figurative extension "put a check on" first recorded 1580s. Related: Reined; reining. To rein up "halt" (1550s) is from the way to make a horse stop by pulling up on the reins.