late 14c., "expression of (popular) discontent or complaint by grumbling," from Old French murmure "murmur, sound of human voices; trouble, argument" (12c.), noun of action from murmurer "to murmur," from Latin murmurare "to murmur, mutter," from murmur (n.) "a hum, muttering, rushing," probably from a PIE reduplicative base *mor-mor, of imitative origin (source also of Sanskrit murmurah "crackling fire," Greek mormyrein "to roar, boil," Lithuanian murmlenti "to murmur").
Meaning "a low sound continuously repeated" (of bees, streams, etc.) is by c. 1400. That of "softly spoken words" is from 1670s. Medical sense of "sound heard in ascultation" is by 1824.