It's interesting to me inasmuch as the existence of such phrases/idioms suggests that the underlying folk metaphysics is 'all things need to be paid for somehow, they do not come free'. Which seems obvious when it comes to things usually in material exchange, such as foodstuffs, clothing, pots etc. etc.. But the fact that we say things such as 'I don't give a dam(n) what you think of me' suggests that also such things as respect, attention, affection come at some kind of cost. Which puts an interesting spin on proclamations of 'equality for all', 'liberty', 'democracy', 'unconditional kindness' and the like. These things simply are not free, they aren't even cheap, as much as their proponents like to believe they are.
When I was a child circa 1950 my mother told me it was a tinkers tool worth less than a penny (almost worthless)
Humor here . . . . . I think you are all wrong! . . . . . in the old US Civil War days (ca. 1864-ish), Rhett Butler, a southern dandy, said to his girlfriend, Frankly (or significant other): "Frankly, my dear, I don't give a damn!" HAHAHA!!! . . . . . yes . . . . I Know!!
That line of dialog was in "Gone with the Wind," a movie made in the 20th century. It's not good scholarship to assume that people actually spoke that way three generations earlier!
The word is neither Hindi nor Urdu, originally. "Dam" is a Bengali/ Bangla word. In Hindi, it's "Bhao". As the English East India Company had their first establishment in Bengal, alltheir officers learned Bengali,, and adopted many Bengali words (like thug, thugee, bungalow) into English, as Hobson-Jobson recorded. Some of these Bengali words are also the same as in Hindi, because most of the words in both languages are derived from Sanskrit. "Dam" means "price". So, "I don't give it a dam" means "I don't pay any price for it" - or "I don't give it any value". Later, as you rightly wrote, this word changed to "damn", in which case the idiom loses its original meaning. If you "don't damn it", you are actually giving a value to it!
The value given is: Not worth the petty swear of a tinker ( a low-class person stereotypically inclined to bad language); not significant enough even to be worth routinely cursing, combined with the pun (the mechanical dam used to contain solder or other hot liquids; intentionally discardable, part of the overhead, even from a cheap, quick, and dirty fix). So maybe more of an extension or elaboration or transferral (a "horizontal transfer" of reference) than a "change" of meaning - an aligned change, contributing to a fairly deep idiomatic expression. Compare "not worth a popcorn fart", or the sophisticated dismissal of a worthless scholarly paper: "This isn't even wrong"; or the probably apocryphal alleged response of boxer Joe Louis to a guy who had punched him in a bar on a dare: "If you ever hit me again, and I find out about it - - - - "
