"Give eye teeth for" is a common idiom used to express the desperate desire for something. Some synonyms for this expression are "crave," "covet," "yearn for," "hanker after," "thirst for," "long for," "pine for," "desire," and "lust after." These words denote intense longing or a strong wish to have something. They convey the idea of feeling a great need for a particular object or experience. Using synonyms for "give eye teeth for" can add variety and nuance to your writing or conversation, allowing you to express yourself in a more diverse and colorful way.