This theory is not my original creation, a few have said it before, but I'm giving my own take on the matter.
I will be answering the Following questions in this theory:
1) How did the Sage come about?
2) Why is the Byakugan so Inferior?
3) Why is the Uzumaki clan a relative of the Senju?
Starting with Kakashi, he once said that the Hyuugas were a very ancestral clan and legend has it the Uchihas and their Sharingan originated from them.
Let's give Kakashi some credit, being as intelligent and knowledgeable as he is, that he wouldn't be voicing legends. As Tobi said, myths have a base on reality. So let's conviniently assume Hyuugas and Uchiha are related somehow.
Next, the villages of Konoha and Uzushio were once allies, that the clans of Senju and Uzumaki are distantly related.
The Uzumakis are known for their incredible longevity and life force, and because they were masters at Fuuinjutsu.
Now, we have that Hyuugas and Uchihas are related, and that Uzumakis and Senjus are related as well. But we know the origins of both Uchiha and Senju clans, the descended from the Rikudou Sennin's two sons respectively. But how come these four clans are related? The Sage only had two sons, right?.
Well, my theory is that both Uzumaki and Hyuuga clans predate the Senjus and Uchihas. And their connections with the latter two goes through the Rikudou Sennin. Assume the Sage was the product of mixing the lineages of both Uzumaki and Hyuuga. It was from his Uzumaki' parentage that the Sage inherited "the body" and through his Hyuuga's parentage inherited "the eyes".
But something is wrong, how come the Rikudou Sennin's inherited his eyes (Rinnegan) from the Hyuugas?, when they possessed another doujutsu: the Byakugan. This time we're going back to Kakashi, and giving him the benefit of the doubt, he said to Tobi that the Rinnegan was a "mutation". So here it is another piece of the theory, the Uzumaki's blood in the Sage mutated his Hyuuga's inheritance into the Rinnegan.
But still something's fishy, both Orochimaru and Kabuto theorised that since the Sharingan comes from the Rinnegan, it can naturally evolve back to it, right? Moreso, no attention was paid to the Byakugan, as magnificent as it is, it seems VERY inferior to both Sharingan and Rinnegan. And let's presume it is inferior because the Byakugan is the first and most primitive doujutsu in existence.
But I believe Orochimaru, Kabuto, Madara and Obito are got the process backwards. Let me illustrate it like this:
A+B=RS---> A-2 & B-2
A= Hyuuga
B = Uzumaki
RS= Rikudo Sennin
A-2= Uchiha
B-2 = Senju
So Madara and Orochimaru realized that to get to point RS (Rinnegan) you must combine S and . And they must be right, but I believe that the original and purest process is getting A and B together. And I call Uchiha and Senju A-2 and B-2 respectively, because I suspect that these two clans are upgraded versions of Hyuugas and Uzumakis through the genetic mixture the Rikudou Sennin was. Being the latter two the original possessors of the "eyes" and the "body".
There's something more, the Rikudou Sennin invented the first Fuuinjutsu to create a jinchuuriki, and that's an Uzumaki trademark. Perhaps, it was the Sage himself who taught the Uzumakis the secrets of Fuuinjutsu, and also taught the Hyuugas to better utilize the Byakugan, besides fathering and training the Uchiha and Senju ancestors.
How come Uzumakis and Hyuugas haven't gotten together ever since?
Well, I think the first reason is that Uzumakis and Hyuugas lived in different countries, the second reason is how extremely protective the Hyuugas had become with the doujutsu and secrets of their clan, and the last reason is that Uzumakis chose to ally, even marry, Senju clansmen in particular.
That is all, this my take on the origins of the Rikudou Sennin, and how he is the dot connecting Hyuugas, Uzumakis, Uchihas and Senjus.
PS: After reading this, do you think the love Hinata (hyuga) feels for Naruto (Uzumaki) is just coincidence? It could be an echo of the love between the parents of the Rikudou Sennin. Who knows, right?
OR
According to this scan from the latest chapter:
The Juubi has the Sha-rinnegan. The only possibility for the sage to weild the rinnegan is if he/his Line of priests were descendants of the Juubi (

) being that we now know Genetic mutation of doujutsu only happenes within the same bloodline