A Dam was a small Indian copper coin. The coin was first introduced by Sher Shah Suri during his rule of India between 1540 and 1545, along with Mohur, the gold coin and Rupiya the silver coin Later on, the Mughal Emperors standardised the coin along with other silver (Rupiya) and gold (Mohur) coins in order to consolidate the monetary system across India.
Watch Your Language lists the coin as one of the possible sources for the English phrase “I don't give a dam[n]″, due to its small worth, but provides other sources as well.
References
Dam (Indian coin) Wikipedia(Text) CC BY-SA