Green’s Dictionary of Slang

button n.4

(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.

[US]D. Pendleton Boston Blitz (1974) 137: They’d just better deliver, that’s all. I’m through playing button-button.
[US]E. Torres After Hours 5: Had one of them hard-water voices. They [i.e. US Mafiosi] all got them. Comes with the button.
[US]N. Pileggi Wiseguy (2001) 181: Tommy was going to be made. He was finally getting his button.
[US]H. Hill Good Fella’s Guide to N.Y. 163: To earn a ‘button’ was to become a made man, an untouchable.
[US]D. Winslow Winter of Frankie Machine (2007) 133: Frank had got his button for that night [...] They had a cheesy‘ceremony’ in the back of a car.

In compounds

button man (n.) (also button guy)

(US) a lower echelon member of a Mafia family.

[US]M. Puzo Godfather 101: It was an important promotion, that to ‘button’ man, one not to be handed out lightly. The man had to be tough and he had to be smart.
[US]E. Torres Carlito’s Way 20: Pete says to this button-guy with him, ‘He’s a fuckin’ nigger’.
[US]J. Wambaugh Glitter Dome (1982) 191: And all we’re trying to do is stop the button men from hitting the mattresses.
[US](con. 1930s) Courtwright & Des Jarlais Addicts Who Survived 188: You had to be a button guy, you had to kill three, four people and be somebody that they look up to.
[UK]J.J. Connolly Layer Cake 273: You wanna start fuckin playin the gangster, start callin on button men, been watchin too many fuckin pictures.
[[US]C. Stella Jimmy Bench-Press 19: There’s guys out there with buttons couldn’t make first team crossing guards].
[US]C. Goffard Snitch Jacket 97: He might [...] have done a few jobs as a freelance hit-man. [...] ‘Christ,’ I said. ‘A buttonman.’.
[US]G. Pelecanos (con. 1972) What It Was 122: Button men from up north [...] Italians.

In phrases

half a button (n.)

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

[US]E. Torres Carlito’s Way 9: Nino, he had half a button, cool head, he’d talk to you.
[US]E. Torres Q&A 157: Some half-a-button guy.