Well I think Rin'negan and Byakugan were around at the same time.
Rin'negan seems to deal with life and death, summoning the 'King of Hell'
Byakugan seems to have a Heaven connection, has many of there jutsu have heaven in them somewhere in them.
Now in Shinto there were two Gods that started everything, Izanami and Izanagi. Izanami was the Goddess of Creation, she and Izanagi then created many things blah blah.
Now we know the Sage of Six Paths created Modern Ninjutsu, but maybe he had help...
Next Izanami died giving birth (the child Kagu-Tsuchi (incarnation of fire) or Ho-Masubi (causer of fire)) There she became the Goddess of Death.
Izanagi then went to the underworld to try and bring her back, but failed, however during his clean rite when he returned, he begot Amaterasu (the sun goddess) from his left eye, Tsukuyomi (the moon god) from his right eye and Susanoo (tempest or storm god) from his nose.
We also know that the Rin'negan has power over Life and Death, much like Izanami, and from these to the world they know of came forth, much like the Sage of Six Paths did with Ninjutsu.
So maybe the Byakugan and Rin'negan come about at the same time, and somehow, the Sharingan is a cross between the two, it would explain why it has aspects of both
It has Time Space abilities, and great Genjutsu, things the Rin'negan has, and it's vision enhancements like the Byakugan.
It also talks about Izanagi 'sealing' Izanami away, maybe something to do with why the Rin'negan is so rare, or what is sealed in the Moon.
