"who didn't give a tinker's dam about him"

Always used as a negative, “not worth a tinker’s dam” is merely a superlative version of “not worth a damn.” Tinkers, the term for traveling tinsmiths who repaired household utensils made of tin, were generally known as particularly colorful employers of swear words. So if something wasn’t worth even a “tinker’s damn,” it was particularly worthless.