Green’s Dictionary of Slang

(US) a nickname for anyone seen as overweight.

1930		J. Lait Put on the Spot 109: Fats Ryan, a crimson-cheeked Irishman.
1953		L. Durst Jives of Dr. Hepcat (1989) 4: From off the end comes ‘Fats’ and his buddy, ‘Mule,’ the duece [sic] that is able and from the righteous school.
1957		‘Ed Lacy’ Lead With Your Left (1958) 11: Fatstuff, you already made too many cracks bout my being young.
1960		(con. 1940s) G. Morrill Dark Sea Running 20: How are your sea legs, Fat Stuff?
1969		C. Himes Blind Man with a Pistol (1971) 156: But Fats didn’t have any white women — he liked boys.
1971		N. Smith Gumshoe (1998) 166: I think my ears are on the blink, Fats.
1990		S. Morgan Homeboy 119: Just can’t get good help these days, huh, Fats?