Wiktionary
n. (context North America slang English) wide-open eyes, as from being surprised or startled.
Wikipedia
Banjo Eyes is a musical based on the play Three Men on a Horse by John Cecil Holm and George Abbott. It has a book by Joseph Quinlan and Izzy Ellinson, music by Vernon Duke, and lyrics by John La Touche and Harold Adamson.
Produced by Albert Lewis and staged by Hassard Short, the Broadway production opened on December 25, 1941 at the Hollywood Theatre, where it ran for 126 performances. The cast included Eddie Cantor, Lionel Stander, and William Johnson.
Although Cantor was known as "Banjo Eyes," the title referred not to his character but to a talking race horse, played in costume by the vaudeville team of Morton and Mayo. In dream sequences, Banjo Eyes would give Cantor's character tips on which horses were going to win different races, but warned him his supposed talent for picking the winners would vanish if he ever placed a bet himself. The book was a very loose adaptation of its source, and the World War II anthem "We Did It Before (And We Can Do It Again)" by Charles Tobias and Cliff Friend was interpolated into the score for no apparent reason other than to stir up patriotism among audience members. Cantor closed the show by singing a medley of his hits in his customary blackface. The show closed when its star suffered a medical emergency.
Usage examples of "banjo eyes".
He rolled his banjo eyes at Laura as he held up a shimmery scantling.