give one's eyeteeth
Also, give one's right arm. Go to any lengths to obtain, as in She'd give her eyeteeth for a mink coat, or He'd give his right arm for a new car. These hyperbolic expressions both allude to something precious, the eyeteeth (or canines) being useful for both biting and chewing and the right arm a virtual necessity for the 90 percent of the population who are right-handed. Both date from the first half of the 1900s, when the first replaced give one's eyes, from the mid-1800s.