Green’s Dictionary of Slang

a euph. for damn n.

c.1894		Adele Levey [perf.] ‘Cherro!’ 🎵 So don't give a hang for the weather / But join hands and all sing together.
1900		Bulletin (Sydney) 2 June 25/3: As I am woefully weak in chronology and topography, I was glad to see by your note [...] that it won’t matter a hang.
1901		H. Lawson ‘That Pretty Girl on the Army’ in Roderick (1972) 481: I don’t suppose it’ll matter a hang in the end.
1903		Bulletin (Sydney) 16 July 32/1: A real beer drinker near me assured me that he never saw a spectacled man who could drink beer worth a hang.
1913		Sport (Adelaide) 1 Mar. 12/1: He don’t care a hang / If here name should be Lang.
1938		Wodonga & Towong Sentinel (Vic.) 7 Oct. 5/3: ‘Geoff! The men can hear you!’ ‘I don’t care a hang if they can!’.
1977		Fidrych & Clark No Big Deal 217: He said, Hey, you talk your baseball, I don’t give a hang.