1
general::
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 righthanded. Both date from the first half of the 1900s, when the first replaced give one's eyes, from the mid-1800s.
American Heritage Idioms