"Eye of Newt" is a phrase used in William Shakespeare's play Macbeth, in a scene where the three witches are throwing a series of ingredients into a cauldron, for the purpose of summoning up shades and spirits to make predictions to Macbeth about his future.
Many of Shakespeare's phrases and coinages have made it into the English language. Of course, "Eye of Newt" might predate Shakespeare's usage of it in Macbeth, but it is from this play that it gained its currency. And like much of Shakespeare, it is not always the best or most relevant passages that have entered our culture the most. The phrase "Eye of Newt", along with "Out Damned Spot!" are probably the most well-known parts of Macbeth, even amongst those who don't know their source.
There are 22 different ingredients mentioned in the passage that "Eye of Newt" comes from. For reasons of clarity and length, "Maw and gulf of the ravined salt-sea shark" never entered our language. Neither, thankfully, did "Liver of Blaspheming Jew". But for whatever reason "Eye of Newt" became the prototypical grotesque ingredient for occult potions, ahead of "Lizard's Leg" or "Gall of Goat". While I could draw some conclusions about this, I will just have to say it is one of the many mysteries that surround the general crystallization of English idioms.