would give your eye teeth
give (one's) eyeteeth for (something)
To go to any length or relinquish anything of value in order to obtain some specific thing in return. Almost always used after the modal verb "would." Oh, I would give my eyeteeth for curly hair like yours. She said she'd give her eyeteeth for a chance to meet the singer in person.
Farlex Dictionary of Idioms. © 2015 Farlex, Inc, all rights reserved.
would give your eye teethBRITISH, INFORMAL
If you would give your eye teeth for something or to do something, you would love to have or do that thing. He's the most exciting man I've ever worked with, and I'd give my eye teeth to do something with him again. I'd give my eye teeth for an opportunity to join that company. Note: A person's eye teeth are their canine teeth, the pointed teeth near the front of their mouth.
Collins COBUILD Idioms Dictionary, 3rd ed. © HarperCollins Publishers 2012