The exact relationship between the noun and verb blab and blabber is difficult to determine. The noun was "[e]xceedingly common in 16th and 17th c.; unusual in literature since c 1750" [OED]. Middle English also had lab (v.) "talk foolishly, let out a secret" (late 14c.), said to be from continental Low Germanic; hence also labster (Middle English labestere, late 13c.) "female gossip, a scold."
early 14c., "to speak," from mouth (n.). Related: Mouthed; mouthing. Old English had muðettan "to blab." In 17c.-18c. especially "to speak pompously or affectedly." Meaning "form the shape of words with the mouth without uttering them" is by 1953.