The word "Horus" was the name of an ancient Egyptian sky god who was symbolized as a falcon. The right eye of the falcon was associated with the sun, called Ra. The eye symbol represents the Peregrine Falcon's eye with a teardrop in it. The left eye sometimes symbolizes the moon and another god, Djehuti (Thoth).
I read this summary of the Egyptian myth about the eye on one website:
"According to the old myth, the rivalling god Seth tore Horus' eye out. Seth was his uncle, who contended with him for the Egyptian throne after he had killed and dismembered his father, Osiris. Thot, the wise moon god and the patron of the sciences and the art of writing, put it patiently back in order and healed it. As an ambiguous symbol, it describes the status of regained soundness."
The eye would often be painted on boats to insure safe travel. Perhaps that's part of why this eye (below) is hanging in the Travel Office at American University in Cairo.
Here's a photo of a section of mural in Maadi near where we live.