late 14c., reume, "watery fluid or humid matter in the eyes, nose, or mouth" (including tears, saliva, mucous discharge from the nostrils), from Old French reume "a head-cold" (13c., Modern French rhume) and directly from Latin rheuma, reuma, from Greek rheuma "discharge from the body, flux; a stream, current, flood, a flowing," literally "that which flows," from rhein "to flow" (from PIE root *sreu- "to flow").
In old medicine it was conceived as draining from the higher to lower parts of the body and causing ailments if out of balance. Also from late 14c. as "a head-cold, catarrh." The -h- was restored in early Modern English.