Green’s Dictionary of Slang

1. a thin slice of bread and butter.

1861		(con. 1840s–50s) H. Mayhew London Labour and London Poor I 52/1: For this sum they can procure a small cup of coffee, and two ‘thin’ (that is to say two thin slices of bread and butter).
1866		Wild Boys of London I 32/1: Two slices of stale buttered thin.

2. (US) ten cents [thin dime].

1925		T.A. Dorgan in Zwilling TAD Lex. (1993) 82: Ha — ha Ole 8-ball got stung — He picked up a man wid a golf suit, packed 3 bags fo half a hour an’ got a thin.
1944		D. Burley Orig. Hbk of Harlem Jive 12: All the stud’s laying down is a deuce of demons, because he don’t spread nothing but thins on the line.
1947		M.H. Boulware Jive and Sl. n.p.: Scratch me a thin ... Give me a dime.