c. 1300, rinsen, rincen, "subject to light washing; wash with water only" (originally in liturgy; from mid-13c. in surname Rinsfet), from Old French reincier (transitive) "to wash, cleanse" (12c., Modern French rincer), probably a dissimilation of recincier, from Vulgar Latin *recentiare "to make fresh, to wash, cleanse with water," from Late Latin recentare "to make fresh," from Latin recens "new, fresh" (see recent). OED says any similarity in form and sense with Old Norse hreinsa is "prob[ably] accidental."
In general use, of bowls, cups, etc., c. 1400; the meaning "wash (laundry) a second time to remove remaining impurities, soap, etc. that may have been left" is by c. 1500. Related: Rinsed; rinsing.
"a light washing; a renewed application of water to remove residue from a former washing," 1837, from rinse (v.). As a hair treatment, by 1928. An earlier noun was Middle English rincinge (c. 1300).