A dark eye is mentioned by Benerro, the High Priest of the red temple in Volantis, in reference to something watching Daenerys Targaryen. It is not known who or what the dark eye is or if it is merely a figure of speech.
A Dance With Dragons
Benerro preaches that Daenerys Targaryen stands in peril and that the dark eye has fallen upon her. After listening to Ser Jorah Mormont's translation, Tyrion Lannister realizes that the red priest speaks of an ancient prophecy which foretells the coming of a hero to deliver the world from darkness. It is unknown if this is the prince that was promised.
The Winds of Winter
This information has thus far been released in a sample chapter for The Winds of Winter, and might therefore not be in finalized form. Keep in mind that the content as described below is still subject to change.
Theon Greyjoy recalls that his uncle, Euron Crow's Eye, uses an eyepatch to hide a black eye shining with malice. Euron has dispatched his brother, Victarion, to retrieve Daenerys with the Iron Fleet.
Tyrion: What is he saying?
Jorah: That Daenerys stands in peril. The dark eye has fallen upon her, and the minions of night are plotting her destruction, praying to their false gods in temples of deceit ... conspiring at betrayal with godless outlanders ...
Moqorro: Others seek Daenerys too.
Tyrion: Have you seen these others in your fires?Moqorro: Only their shadows. One most of all. A tall and twisted thing with one black eye and ten long arms, sailing on a sea of blood.