Possibly from Medieval Latin *carminium, from Arabic قِرْمِز (qirmiz, “crimson, kermes”) (from Sanskrit कृमिज (kṛmija, “produced by worms”), from कृमि (kṛ́mi, “worm, insect”)), plus or with influence from Latin minium.
Or, from a contraction of Old French carmesin, from Spanish carmesí.