give one's right arm, to

give one's right arm (for someone or something)

 and give one's eyeteeth (for someone or something)
Fig. to be willing to give something of great value for someone or something. I'd give my right arm for a nice cool drink. I'd give my eyeteeth to be there.
See also: arm, give, right

give one's right arm, to

To go to any lengths. Also put as cut off one’s right arm for, this term expresses a major sacrifice for all but the left-handed, perhaps even more so than to give one’s eyeteeth. An Americanism from the twentieth century, it presumably was well known by the time Robert G. Dean used it in Layoff (1942): “He’d cut off his right arm for her, as the saying goes.”
See also: give, right