Give one’s eye-teeth (for something)
To give one’s eye-teeth for something or other means that one would go to extreme lengths, even to the extent of giving up one’s eye-teeth, in order to achieve or gain something. Eye-teeth are the ‘canine’ teeth in the human upper jaw, so-called because such teeth are very prominent in dogs. They are called ‘eye-teeth’ in humans because of their position directly under the eyes and were first called ‘eye-teeth’ from the late 1500s, according to the OED. The notion of parting with one’s eye-teeth in order to gain something is first attested from the late 18th century, when Town & Country Magazine in 1779 reported that a certain publisher would “as soon part with his eye teeth as a guinea”. The change from this to ‘giving one’s eye-teeth for something’ is first recorded from the early 19th century.