No one can spin a better yarn about the eyes as the ancient Egyptians. In the Egyptian mythology, the right eye, aka the Eye of Re (or Ra), symbolizes the Sun. And the left eye, aka the Eye of Horus, is the Moon.
The Eye of Re has a mind of its own and frequently wanders around by itself. On one of the fun-filled outings, it refuses to return, so Re (the father of all gods) sends Shu and Tefnut, two underlings, to retrieve it. The eye stubbornly refuses. And in the ensuing struggle, the eye sheds tears and from which, men emerge.Another story has it differently, Re actually sends Thoth to fetch it back. And upon returning, the eye discovers that it has been replaced by another eye and becomes quite indignant. To pacify it, Re places the right eye in the shape of the uraeus serpent on his own brow - to show that the eye is now the ruler of the world. This serpent is worn by all Egyptian Pharaohs on their headgears. The right eye later becomes an entity and its association with many goddesses is of course the basis of many other stories.
The story of the left eye, the Moon, is even more complex. 
The lunar cycle actually represents the eternal battle between Horus and Seth. They are fighting for the inheritance of Horus's father, Osiris. Seth steals the eye and damages it - cuts it up into six pieces. Thoth, with the help of other gods, puts the eye back together on the 6th lunar day. The healed eye is known as Wadjet. It symbolizes a re-established order after disturbances. The lunar cycle is essentially a description of how the eye is injured and repaired. The latter performed by 14 gods.
For the mere mortals such as you and I, the two eyes also behave somewhat differently - in the form of ocular dominance. About 2/3 of the population sight with the right eye. Ocular dominance seems to correlate with the handedness, i.e., a right-handed person sights with the right eye. However, there are always exceptions; you can be right-handed but sight with the left eye and so on. The reason is simple, each retina is controlled by both hemispheres of the brain, whereas each limb by only one side.
Ocular dominance is actually quite interesting and practical at the same time. It is often why a patient notices a change in vision (i.e., when the dominant eye is seeing less well than the fellow eye) and seeks help. In monovision contact lens fitting, the first attempt is always to fit the dominant eye for distant vision and the fellow eye for reading. Sometimes, this does not work out and the patient actually does better with the dominant eye for reading. In monovision laser vision correction, the dominant eye is also selected for distant vision. You are of course stuck if this arrangement does not work.
Being very capable archers, the ancient Egyptians must have queried their eye doctors why some sighted with one eye vs the other. The mythology of the eyes, on the other hand, is a true testament to their vivid imagination.
Evidence of eye surgery in ancient Egypt? See the lower eyelid of the right eye in this mummy portrait: 
An attempt at repairing entropion, it seems.
