Green’s Dictionary of Slang

(US) a person, usu. criminal, of little or no importance or influence.

1979		E. Torres After Hours 28: What can he give us? [...] Couple of flyweights out of Jocks.
1996		(con. 1949) G. Pelecanos Big Blowdown (1999) 184: The two of them would have had a good laugh now at the expense of this flyweight.