(Also "I don't give a tinker's cuss")
As with many good colloquialisms this is falling out of favour for a number of reasons, not least of which is that most people don't know what a tinker is (or was), even among those who know what a "cuss" is (and know that it isn't an American word!).
To an Englishman, a tinker's everyday language is so riddled with curse words that the value of one is extremely low. That's what makes this an elegant expression; it calls to mind cussing, which allows the user to express annoyance, but it very specifically means "valueless" and therefore at the same time devalues the very cussing that it calls to mind. Which sort of makes it self-excusing. I love that.