"Not give a tinker's damn" is a colorful idiom used to express an extreme lack of interest or concern about something. There are many synonyms and variations that can be used to convey similar meaning such as "not give a rat's ass," "not give a hoot," "not care a fig," "not give a damn," "not care a bit," "not give two cents," "not give two hoots," or "not give two shakes." All of these phrases imply a disinterest or indifference towards something and convey a similar message to the original idiom. These colorful phrases add personality and emphasis to a conversation, and convey a certain attitude of indifference or apathy towards the topic at hand.