The phrase "would give your eye teeth for something/to do something" is an idiom that means someone is willing to give up something of great value or importance to obtain something they desire greatly. Synonyms for this phrase might include offering a prized possession, sacrificing something dear, putting something valuable on the line, or going to great lengths to achieve a goal. Other comparable expressions include "pay through the nose," "jump through hoops," or "bend over backward." These phrases all connote a high degree of motivation, passion, and determination to accomplish a task or obtain something valuable.