Green’s Dictionary of Slang

(US) large, wide-open eyes; thus a person with such eyes; also in direct address.

1901		Hawaiian Gaz. (HI) 5 Apr. 5/1: They were off in a corner by themselves making banjo eyes and telling each other [etc].
1902		Ade Girl Proposition 101: The latest variety of New Woman resents the Suggestion that she is a Soft Mark for the curbstone Masher who stands in front of Cigar Stores and Works the Banjo Eye.
1928		F. Willard ‘Moon Mullins’ [comic strip] What, what day would be more convenient for you, banjo-eyes? [...] Yes, that banjoeyed Mullins fellow answered the door.
1961		(con. early 1950s) J. Peacock Valhalla 271: What’re you lookin’ at, banjo eyes?
2001		T. Fontana ‘Orpheus Descending’ Oz ser. 4 ep. 14 [TV script] I’ll stick it in myself. Right between his banjo eyes.
2007		N. McCall Them (2008) 37: A short, bald, bow-spined man with banjo eyes.