Green’s Dictionary of Slang

goo-goo v.

(US) to make eyes at someone.

1902		Ade Forty Modern Fables 111: She put her Chin on his shoulder and Goo-Gooed him and he lost the Power of Speech.
1908		H. Green Maison De Shine 190: ‘Never mind turnin’ ’round an’ goo-gooin’ Birdie Trippit,’ warned Mrs. La Mutte.
1914		E. O’Neill The Movie Man in Ten ‘Lost’ Plays (1995) 190: Some one of these Mexican dolls you’re goo-gooing at will carve her initials on your back with the breadknife some one of these days.
1935		M. Fulcher ‘Believe Me’ in Afro-American (Baltimore, MD) 9 Mar. 12/7: It was the Howard U. co-eds who goo-gooed at Chuck Richards.

In derivatives

goo-gooer (n.)

(Aus.) a lover.

1905

Sun. Times (Perth) 2 July 1/1: The goo-gooers ‘from the goldfields’ didn’t come from any remoter camp than Claremont [and] the honeymooning is carried on without the knowledge of a husband.