Is it possible for a Byakugan user to awaken the Rinnegan? Well... the title says it!

I ask this because, supposedly, Hyuuga and Uchiha are descendants of the Rikudou's eldest son (at least I am under that impression). In that way, does the Byakugan have any other special techniques that can be awakened (such as Tsukuyomi, Susano'o and Amaterasu in the Sharingan), and can it (ultimately) awaken the Rinnegan?

So far we have not seen techniques that derive from the Byakugan itself, but rather techniques that are possible because of it: the whole Hyuuga fighting style comes from the use of Byakugan, but it is not related to it in the same way that every Mangekyou technique is related to the Sharingan. They were only able to develop/and use it properly due to them having the Byakugan, but (again) it does not relate to it in the same way the Mangekyou relates to the Sharingan.

To sum up:
Is it possible for a Byakugan user to awaken a Rinnegan?
Unlikely.

While it is true that Kakashi originally said that the Uchiha clan (And the Sharingan) is a branch off the Hyuuga clan (and the Byakugan), it was later revealed that it wasn't the case And even though Kaguya Otsutsuki was said to have the Byakugan, it is currently presumed that the two bloodlines are separate and there's no actual connection between the two eye techniques..

No Byakugan upgrades were ever discussed/revealed, like you said, the Hyuuga specific techniques are only the Byakugan's potential being utilized.

SPOILER!


   Moreover, it was revealed that the way to awaken the Rinnegan would be to combine Indra's and Ashura's chakra, like Madara (who had Indra's chakra) implanted the flesh of Hashirama (who had Ashura's chakra), to bring out the Sage of Six Paths' chakra.

 So theoretically, if Indra were to posses a Hyuuga clan member, and the identity of Ashura would be known, it may be possible. That, however, never happened and is not likely to happen in the future.