It's not just one eye - it's one BIG eye.
The Cyclops is a large hominid species that has evolved as a specialized nomadic shepherd. In order to decide where to lead its flock next, being able to see long distances - and as such, where it might find the best pastures and smallest amount of competition from other cyclopes or wild flocks - is more important than depth perception, since it doesn't need to hunt anyway. Large eyes are also good for night vision, which a shepherd will need to keep watch for nocturnal predators.
The animal with the most comparable adaptation is the ostrich, a bird famous for having eyes larger than its brain. The cyclops, however, is an intelligent species. There is a lot of intraspecific competition between them - when deciding where to lead their flock, they also have to anticipate where other cyclopes are likely to lead their flocks, in order to avoid unwanted competition. As such, they need to be capable of long-term planning coupled with a strong theory of mind. They may also have to keep track of agreements or truces regarding particular areas of land. As such, it needs at least one big, long-distance eye, and a big brain.
There isn't enough room in its head for two oversized eyes and a big brain. So the best trade-off is to lose one eye.
One might think that losing one eye is problematic for a shepherd because it reduces peripheral vision and therefore could miss sight of a predator, but remember that grazing animals normally keep their heads low to the ground in order to graze, and as such need the ability to spot predators out of the corner of their eyes quickly. A cyclops, by contrast, can stand guard on a nearby hill and slowly rotate to get a clear view of the landscape, including all potential predators, for miles around - it has nothing better to do. Also, if a predator gets through, the cyclops doesn't have to be concerned about its own life, the worst that could happen is that it will lose a few sheep. So good distance vision and intelligence trump peripheral vision in this case.
The early cyclopes were likely binocular hominid shepherds that started to develop one of their eyes to be bigger than the other, using their larger eye to see further away. Over time, the smaller, less-used eye atrophied and eventually disappeared, and the main eye became more centralized.
This lifestyle also explains the cyclops' large size - height for seeing farther, strength for fighting off predators and wrangling its flock, and as a shepherd they have easy access to meat and protein. Of course the colossal size they are typically depicted with is an exaggeration, but maybe nine or ten feet tall with a wide, squat shape could be fairly realistic.