Give your eye teeth for something/to do something
Definition 1 of 1
Pro Tip 1/3Intensity
The phrase denotes an intense desire or willingness to sacrifice greatly for something. 
She would give her eye teeth to meet her favorite author.
Pro Tip 2/3Metaphorical
Used metaphorically; no one actually offers their teeth for exchange. 
I'd give my eye teeth for a vacation in the Bahamas right now.
Pro Tip 3/3Hyperbolic
It's hyperbole, which means it's an exaggeration to make a point, not to be taken literally. 
He'd give his eye teeth for a promotion, he's worked so hard.