It's not a well known fact that the Hyuuga didn't always have x-ray eyes like they do now. In fact, during the time of the Nidaime, the Hyuuga were a normal ninja clan. An extremely homosexual and perverted ninja clan, but they were normal in abilities.
While all of them were incredibly homosexual and perverted, there was one, Jiraiya Hyuuga (the sennin we know and love was incidentally named after this pervert, but he went a different direction and is a different story) whose pervertedness and homosexuality made the rest of his family seem like innocent heterosexual young boys.
Jiraiya was constantly inventing new perverted things, like Kage Bunshin pants, that would disappear with a little pat (quite a few males were unfortunate to fall to this), Kage Bunshin Underwear, the Wonder Bra, and Vaseline. While these useless inventions would eventually get left behind in history, he created one perverted jutsu that would alter his clan forever. The Byakugan, was Jiraiya's genius all purpose peeking technique that could see through walls, clothing, and if you so chose to, you could even see the inner coils (but being perverts, the Hyuuga never did that, after all, how useless is being able to see the inner coils when you can stare at men without their clothes on?).
Jiraiya was so enamored with his new technique that he left it on for an entire month, walking around looking through walls of hot springs and through the clothing of males of all ages. By the time he turned it off, his eyes were permanently white, but obviously that's a small price to pay for unlimited peeking. In fact, when he revealed and taught it to the rest of the clan, he was hailed instantly as a Hyuuga hero, and they rallied to have him made as the Gaykage. But the straight village quickly put them down.
All the Hyuuga constantly using the technique that eventually their eyes all turned white as well, and the technique was passed down from generation to generation. As it passed down to present day, the purpose of the Byakugan was skewed from all purpose peeking technique to scary looking eyes of doom that could see your inner coils.
And that is how the Eyes of the Hyuuga, the Byakugan, came to be.