Green’s Dictionary of Slang

1. a cooked egg, fried or dropped.

1886		N.Y. Dispatch 1 Aug. 7/6: ‘Would you call dropped eggs alone “soldiers?”’ ‘Oh, no, that’s “sleeve buttons”.’.
1901		Times (Wash., DC) 28 Apr. 4/6: Eggs, both sides: ‘One pair sleeve buttons, right and left’; or, ‘one pair sleeve buttons, slapped in the face’. Eggs, one side: ‘One pair sleeve buttons showing the moon’.
1909		I.L. Nascher Wretches of Povertyville 93: The diner gives his order to the waiter, who translates it into the expressive slang of the Bowery as he shouts into the kitchen [...] ‘sleeve-buttons’ or two ‘sunny side up’ for two fried eggs.

2. codfish balls; fishcakes.

1883		Hawk Eye (Burlington, IA) 8 Nov. 3/3: ‘Sleeve buttons,’ [...] is the New York restaurant waiter’s slang for codfish balls.
1888		N.Y. Herald 1 Apr. 9/6: During his stay in the restaurant the reporter learned several things he never knew [...] That ‘sleeve buttons’ meant fish cakes.