What do you think of Weekley's observations that other such similar expressions [don't give tuppence/brass farthing/button/straw], all relate to low value coins or valueless stuff?
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.