Now to start this off I want to point out something that Kakashi said:
This is something people have talked about for a very long time as it would mean that the Uchiha are descendants of the Hyuga. But the thing is, we already know that they are descendants of the Sage, and thus for it to be true the Sage must be a Hyuga. That was the basic idea behind people calling the Sage a Hyuga.
The only problem with this is that we recently found out that the Sage is infact not a Hyuga but part of a completely different clan. So then what is up with Kakashi's statement?
Before I get to that I want to point out that there is more than likely a connection between the Byakugan and the Sharingan. What are the odds that 2 of the 3 Great Doujutsu (Sharingan, Byakugan, Rinnegan) are related and the 3rd is just some random occurance... No I believe there is some relation. We just need to find out what that relation is.
Back to what I was saying, with Kaksahi's statement there are 2 possibilities regarding the Byakugans relation to the Sharingan.
1) That it really did come before the Sharingan, which means the Sage was in someway a Hyuga. Either he too was a descendant of the Hyuga, or his clan later became the Hyuga, or his dad was a Hyuga, etc... But I dont like this as it seems nonconclusive and evasive. *We find out the guy wasnt a hyuga, ok well he parent must have been, find out parent wasnt a hyuga, ok his grand parent must have been, etc...*
2) Kakashi was wrong and the Hyuga came out of the Uchiha... Well this is somewhat obvious but lets entertain this idea a little bit..