A possible etymology for "give a dam(n)"

Take the entry for the Indian word dam. The dictionary defines it as: "Originally an actual copper coin. Damri is a common enough expression for the infinitesimal in coin, and one has often heard a Briton in India say: 'No, I won't give a dumree!' with but a vague notion what a damri meant."

That is the etymology of dam. But Yule and Burnell have more to say.

"And this leads to the suggestion that a like expression, often heard from coarse talkers in England as well as in India, originated in the latter country, and that whatever profanity there may be in the animus, there is none in the etymology, when such an one blurts out 'I don't care a dam!' in other words, 'I don't care a brass farthing!'"

From an article at the BBC discussing English words which originated in India.