Green’s Dictionary of Slang

button n.4

[1970s+] (US Und.) the status of being in a US Mafia ‘family’; usu. as earn/get one’s button v., to be made a member of the Mafia.

In compounds

button man (n.) (also button guy)

[1960s+] (US) a lower echelon member of a Mafia family.

In phrases

half a button (n.)

[1970s] (US Und.) a criminal who is on the fringe of full membership of the US Mafia; also attrib.