late 15c., "one who splits," agent noun from cleave (v.1). Originally "one who splits boards with a wedge instead of sawing;" attested as part of a surname from mid-14c. Meaning "butcher's chopper" is from mid-15c.
This last ["Marrowbones and Cleaver"] is a sign in Fetter Lane, originating from a custom, now rapidly dying away, of the butcher boys serenading newly married couples with these professional instruments. Formerly, the band would consist of four cleavers, each of a different tone, or, if complete, of eight, and by beating their marrowbones skilfully against these, they obtained a sort of music somewhat after the fashion of indifferent bell-ringing. When well performed, however, and heard from a proper distance, it was not altogether unpleasant. ... The butchers of Clare market had the reputation of being the best performers. ... This music was once so common that Tom Killigrew called it the national instrument of England. [Larwood & Hotten, "The History of Signboards from the Earliest Times to the Present Day," London, 1867]