Green’s Dictionary of Slang

(US) an affectionate tickling under the chin or caressing, usu. directed at an infant; thus sexual caressing.

1953		S. Bellow Augie March (1996) 86: Don’t play gidgy with your little girlfriends.
1969		Rowan and Martin’s Laugh-In [NBC-TV] What would you say to a little gidgy? [HDAS].