Green’s Dictionary of Slang

damn n.

also dam
[the orig. ety. is based on the dam, a low value Indian coin, but the widely assumed link is to the oath damn! excl.; the inclusion of ‘twopenny’ was apparently popularized by the Duke of Wellington (1769–1852)]

a minuscule or virtually non-existent amount; often in not give a damn v.

[UK]Grose Classical Dict. of the Vulgar Tongue n.p.: Dam, a small Indian coin, mentioned in the Gentoo code of laws: hence etymologists may, if they please, derive the common expression, I do not care a dam, i.e. I do not care half a farthing for it.
[UK]C.K. Sharpe letter Correspondence (1888) I 211: I come home cussing myself for going to see an experiment ‘not three dam’s worth’ .
[UK]‘One of the Fancy’ Tom Crib’s Memorial to Congress 4: And nought but your Honours worth wasting a d--n on.
[Aus]Bulletin (Sydney) 11 July 7/2: Now, poor old George Loder’s remains lie uncared for in the Adelaide (S.A.) cemetery, and from the time of his death to the present date nobody seems to have sent one single, solitary dam’s-worth of enquiry on his track.
[UK]A. Morrison Tales of Mean Streets (1983) 117: You ’ave to shed blood in a revolution, an’ a few odd lives more or less don’t matter — not a single damn.
[US]S.J. Perelman ‘If An In-Law Meet An Outlaw’ in Keep It Crisp 158: He’ll never amount to a damn.
[Ire]F. O’Connor An Only Child (1970) 103: As a protector of the weak, I was never worth a damn.

In phrases

give a damn (v.)

see separate entry.

not give a damn (v.)

see separate entry.

not matter a damn (v.)

see separate entry.

not worth a damn

see separate entry.