In a number of series, magic is eye-based, where the eyes either fuel the magic or are the conduit for it. Examples of this are Fai in Tsubasa Reservoir Chronicle and the Uchihas and Hyuugas in Naruto, as well as a few other families in Naruto. Is this based on a specific tradition or have a specific cultural history. If not, where was the first time this appeared in anime/manga?
First off where was the first time this appeared in anime/manga? I believe that would be Basilisk, which is based on a novel The Kouga Ninja Scrolls wrote in 1958.
As I could not find a definite source saying were it came from, all I say here will mainly be speculation.
There also is a statement The eyes are windows to the soul. A lot of anime eye powers are based on a soul or other form of power. One example would be Naruto:
Chakra is essential to even the most basic technique; it is the moulding of the physical energy present in every cell of the body and the spiritual energy source
Another statement that's a bit more from the scientific corner is "The eyes and mind are still one big mystery"
Even in science, it is one big mystery what your brain is exactly used for (at least some spots of the brain). As the eyes are directly connected to the brain, the scientists believe the eyes could be influenced by these unactivated parts of the brain
The 10% of brain myth is the widely perpetuated urban legend that most or all humans only make use of 20%, 10% or some other small percentage of their brains. It has been misattributed to people including Albert Einstein. By association, it is suggested that a person may harness this unused potential and increase intelligence. Though factors of intelligence can increase with training, the popular notion that large parts of the brain remain unused, and could subsequently be "activated", rests more in popular folklore than scientific theory. Though mysteries regarding brain function remain—e.g. memory, consciousness — the physiology of brain mapping suggests that most if not all areas of the brain have a function.
One possible origin is the reserve energy theories by Harvard psychologists William James and Boris Sidis in the 1890s who tested the theory in the accelerated raising of child prodigy William Sidis to effect an adulthood IQ of 250–300; thus William James told audiences that people only meet a fraction of their full mental potential, which is a plausible claim
This is also a reason why it could be used in anime/manga. As a lot of manga/anime contain fantasy objects fague theories like this provide for perfect creative potentials.
The eyes are also a good placement for special abilities. Eyes are almost always in the screen as most of an anime expressions will be shown through their facial responses (big eyes over exaggerating smiles etc etc). As a viewer, your main attention will also be placed on these body parts.
As I said this is mainly my own point of view, and will be looking forward to more answers/information.
Magic coming the eyes are part of Magic Eye Trope which can be linked to the origin of having Psychic Powers which already started in 1870, which I think is also the origin of characters having source of power or magic from their eyes.
A character's eye has great supernatural power. Usually, the eye grants the user power over perception, either the target's (illusion, mind control) or the user's (telepathy, premonitions, clairvoyance, etc). In most cases, direct eye contact or at the very least line-of-sight is required. Other restrictions may apply.