Ok hear is my speal on it so bear with me.
The Hyuga clan in part I had two things said about it.
1. It is Konoha's oldest clan...this was said as a fact by Kakashi
2. The Sharingan derives from the Byakugan...this was said as a rumor by Kakashi
I think Kakashi is flat out wrong. The Sharingan and Byakugan developed independantly of one antother, but the both have the same source: The Elder Son.
The key to this whole theory is the fact that it was stated that the DECENDANTS of the Elder Son formed the Uchiha Clan. This means that there was some interval between the Elder Son and the Clan's start. Because of this it is entirely possible to consider this may have taken several generations. Here is where the Hyuga and Uchiha come into play
I believe that the Hyuga derrived from the Elder Son; however, their lineage came from a more branched off family. For example, a female decendant of the Elder Son marries a random gentleman....they have a daughter, and then she marries yet another random gentleman. This process may have taken away some of the Elder Son's Dojutsu leaving the blank white eyes. Eventually after maybee one or two branch offs in this fashion the Hyuga Clan formed, and they formed as an OFFICIAL clan before the Uchiha.
The Uchiha on the other hand, came from a more linear or less diluted/branched off family. One could say maybee a line of Sons born and so the family kept more control over a direct descent (sadly this sounds sexist, but in fuedal history which the story has some ties to, Women were regulated and viewed behind men in Socio-Political Societal structures...No offense ladies, I apologize). So eventually, some time after the Hyuga formed their clan, the Uchiha independantly of the Hyuga, founded their own clan.
Now, because they share a common ancestor it is likely the families kept some form, no matter how minute, of contact, and thus, when the time period of the Clan Wars came (a historical reference to the Warring States Period in Japan), The Uchiha and Senju were rivals, but it is logical to assume that they had other clans as allies. So with familiar ties, the Hyuga would have allied themeselves with their distantly related Uchiha.
When Konoha was established through the alliance, naturally the other clans allied to the Senju and Uchiha would have been in this pact as well. Thus, the Hyuga joined into the village at the same time as the Uchiha and Senju. Since they formed before these two clans, the Hyuga are the Oldest Clan in Konoha. Because of the distant relations and the Clan alliance in the past, it was rumored the Sharingan came from the Byakugan......
Thus, Hyuga are the oldest clan of Konoha, and they are related to the Uchiha.
Thank you for your time!