mid-15c., blabben, "to talk idly and foolishly, talk too much," apparently from Middle English noun blabbe "one who does not control his tongue" (late 13c.), which is probably echoic. Related: Blabbed; blabbing. From c. 1600 as "to talk indiscreetly."
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."