The first question is why they are called eye teeth? The upper canine teeth, the relatively long pointed teeth, are said to be called that because they are positioned directly under the eyes (weird!). The second question is why would someone say they would give up their eye teeth for something they value more? The expression has been around for centuries, and World Wide Words explains losing them “…would cause one to be severely hampered…” Thus, that other thing you want must mean a great deal to you if you would be willing to give up your eye teeth to get it.