The word eyetooth, which is of English origin, first appears in print in 1580. However, it was a while before it was used as the fulcrum for various expressions. In fact, to give one's eyeteeth for something was the last expression to come into play. It's first recorded in Somerset Maugham's Cakes and Ale, published in 1930: "He'd give his eye-teeth to have written a book half as good." Before that, expressions using eyeteeth related more to becoming an adult, and to acquiring the power and authority that comes with maturation. To cut one's eyeteeth, for example, 'to acquire wisdom and become worldly', first appeared in the 19th century. The Encyclopedia of Word and Phrase Origins says this expression derives from the fact that "the permanent set [of canines] is acquired when a child is passing into young adulthood. It is usually said in the negative, as in 'he hasn't cut his eyeteeth yet'." In the 1700s, eyeteeth referred mostly to the cuspids of dogs, and so, the EWPO speculates, "the phrase may have been suggested by the fact that fighting dogs were considered dangerous to handle when they developed their eyeteeth."
There's more. To draw one's eyeteeth means 'to take the conceit out of one'. To have one's eyeteeth means 'to be wide awake'. (Thomas Dyche, in the 1740 edition of his A New General English Dictionary defines eye-teeth as 'quickness or sharpness of understanding and parts'.) Two hundred and fifty years later, a search of the last twenty years of The New York Times reveals that of all the instances where eyeteeth is used in an expression, the overwhelming use is to give one's eyeteeth for something. Gloating over The National Gallery of Ireland's Caravaggio, for example, Director Raymond Keaveney told the Times in 1990 that "museum directors all over the world would give their eyeteeth to have a painting like this."
It is ridiculous to set a detective story in New York City. New York City is itself a detective story...
AGATHA CHRISTIE, Life magazine, May 14, 1956