I'll start off by saying that the Hyuuga clans Symbol is the Ying-Yang symbol. Their byakugan is without a pupil, and is coloured white, which, through activation, results in a more distinct able pupil.
This next picture was seen during Obito & Konan's battle. The picture depicts Tomoes, representing the Uchiha, circles or ripples, representing the Rinnegan, and a Ying-Yang symbol in the middle, representing the Hyuuga. Not only does this hint at the Byakugan being related to the Rennigan and Sharingan, but it emphasizes my point of the Byakugan being part of the other eyes, as the Ying-Yang symbol is the pupil of the eye in the picture below.
Other things that hint at the Byakugan's relation to the other two Dojutsu are, firstly, Tobi's War-time Mask. His mask seems to represent and/or resemble the Juubi's single eye and/or the three dojutsus as one.
As can be seen, Tobi's mask clearly has all three attributes of the three great Dojutsu. It has the ripples of the Rennigan, the Tomoes of the Sharingan, and is coloured white as is the Byakugan. Now, think of this for a moment. Why wasn’t Tobi’s mask colored orange like all his previous masks were before this one? Why is this the only mask of which he’s worn that bares resemblance to the Dojustsus? Tobi’s mask is one of the largest hints of the three eyes being connected to one another in some way.
Ounce again, this above picture, which depicts the moon, of which it’s colour is white, has Tomoes and ripples on it, which ounce again leads me to believe that the Juubi’s eye colour is in fact, not purple or red as depicted in the Anime, but White, as is the Byakugan.
The only eye techniques that exist in the entire (Canon) series are the Byakugan (Mangekyo) Sharingan, Elder Son’s eyes, and the Rennigan. We already know that the Elder Son’s eyes came from the Rennigan, and we know that the Sharingan is capable of developing into the Rennigan, thus making these 3 eyes related to one another. But wait, what about the Byakugan?
Honestly, I find it absolutely illogical for the Byakugan to not be related to the other Dojutsus in some way, as it would seem completely bizarre for the Byakugan to just be there
with no explanation what so ever as to how it came to be.
During Hinata and Neji’s battle in the Chunnin exams, after activating their Byakugans, Kakashi states that there are rumors that the Uchiha’s Sharingan is possibly an off-shoot of the Hyuga’s Byakugan. Kakashi even goes on to state that the Uchiha clan itself possibly stems from the Hyuuga clan. Of course, this is but a rumor, however all rumors have an origin.
Kakashi also states, as a fact, that the Hyuuga are the oldest clan of Konoha. But how can that be if the village itself was formed by the Senju & Uchiha? My theory is that the Hyuuga’s probably owned or lived on the land that is now Konoha, which goes to show that the Hyuuga do indeed pass the Uchiha in their birth/formation, which leads me to believe the Byakugan to be a Dojutsu that came from the Elder Son’s eyes, which may also explain the two eyes’ similar colours and appearance.
If noticed, the Elder Son’s eyes resemble a mixture of the Byakugan and Renningan, which might imply the Byakugan came before the Sharingan, which may explain where the rumor Kakashi mentioned during the Chunnin Exams came from.
All three eye techniques also have similar abilities with each other. Byakugan and Sharingan share the ability of seeing chakra & better reflexes, while the Byakugan and Renningan share the ability of extended vision and being capable of seeing through Barriers, and the Sharingan and Renningan share the ability of tailed beast manipulation as well as controlling demonic powers.
This next one is REALLY good. In a recent interview with Masashi Kishimoto, he was asked what we could expect in for Naruto in 2013, and he replied that we could expect to see Hyuuga development, among several other things. And since the manga currently has Sasuke talking the founders of the Leaf, Hashirama & Tobirama, not to mention the All-Knowing Orochimaru being there to shed some light, we may find out more about the Hyuuga & Byakugan’s past and origins.
Kishimoto always seems to make parallels. Such as Naruto & Sasuke paralleling Hashirama & Tobirama/Madara, or MinaKushi paralleling NaruHina, or the Legendary Sannin paralleling Team 7, which leads me to my next point. It’s clear that the Senju & Uzumaki are related, so what could be the parallel of the Uzumaki for the Uchiha? The Hyuuga’s easily fit this role of being the Uchiha’s “Uzumakis”.
the Hyuuga are usually seen with a blue aura, the Uchiha are seen with a red aura.
Another theory I have is that the Byakugan does indeed have a second state that can be achieved, but in order to achieve it, you must be pure, to be someone like Hinata, while the Uchiha have to be Impure, since the Uchihas' method of gaining more eye power involves stealing & killing, as well as deep despair. This might explain why there are not any known Hyuuga's with an enhanced Byakugan, as it is harder to be pure than impure. I believe this 2nd state or level to be the spiral eyes of the Sage's Elder Son, which would also represent her affinity with Naruto, who is of the Uzumaki, who have a spiral as their symbol.
Also, Hyuugas have white eyes, while the Uchihas have black, which are the colours of the Ying-Yang sign, further hinting at their relationship and how the Byakugan is a pure eye & the Sharingan is an impure eye. Further more, while Hyugas are ussually depicted with a blue aura, Uchihas are seen with a red one instead.
But then, why is the Byakugan "pure" while the Sharingan is "impure"? Well, my answer to that goes something like this. When the Elder son inherited his father's eyes, he also got some of the Juubi's chakra inside him some how, but it remained dormaint and didn't affect him much. But when his father died AND
chose his younger brother instead of him to be his succesor, he felt hate, jealousy, sadness, envy and anger , which caused the Juubi's sleeping chakra inside him to awaken and cause his white eyes to turn red.
The Elder son then had two children of his own, one with his original white eyes, and the other with his Juubi-influenced eyes, thus eventually giving birth to the Byakugan and Hyuga & Sharingan and Uchiha.
The cloud GREATLY desired the Byakugan for their own keeping going as far as to even kidnap a little Hinata. So, why not go after the Sharingan? Actually, now that I think about it, it seems like more non-Konohaians know about the Sharingan than they know about the Byakugan, as seen by Zabuza. It’s seems clear to me that the main house has been hiding secrets of the Byakugan from everyone for a long time, why else would they take the extra precaution to give their Candet branch family members the “Caged Bird” Cursed seals?
There is also the pairing game that the author has going on. A Hyuuga (related to one of the main clans and, on top of it, of the main branch family) and an Uzumaki (related to the other) getting together should produce something interesting, should it not? Both of them being descendants of the Sage, one of which we know nearly nothing at all but her being a pure-blood and the other one showing clear evidence of having an affinity with the Sage's personality and aptitudes.
Also, with Hinata now seemingly playing a larger role, the Byakugan might as well play a larger role too. I just can’t see her disappearing into the back ground after this. She’s clearly been elevated into a higher possession in terms of a character. I’m not just saying this as a Hinata fan, but with Neji dead, there is no other actual Hyuuga character aside from Hinata who could be used to further explain the Byakugan‘s past and (possibly) it‘s new abilities.