Tyrion is actually a child of the Mad King (House Targaryen) and his mother (House Lannister) which gives him as much claim to the throne as Jon Snow/Aegon Targaryen (child of House Stark and House Targaryen). Too many references have been given out in the previous seven seasons for him not to be potentially a Targaryen. When he killed Tywin Lannister, Tywin claims "You're no son of mine." Maybe metaphorical? Maybe literal? In an earlier scene Tywin also states he only took care of Tyrion because "he couldn't prove he wasn't a Lannister." Maybe there was some controversy or Tywin was actually related to his wife (cousin or something which would fit right into the GoT universe). Plus, outside of Jon Snow and Daenerys Targaryen, Tyrion is the only one that's messed with the dragons and not gotten eaten.
My theory is Tyrion and Jon Snow are related via House Targaryen. Which is also why Tywin (other than her having boobs and a great backside) thinks highly of Daenerys. Which is also why he bonded with Jon Snow early in Season 1. There's a connection there by blood between the three characters.
Now, the wild card.... Arya (House Stark) and Gendry (House Baratheon) got together in Episode 802. Arya gets preggers from that encounter and now you have another child with the potential to lay claim to the throne since you now have two powerful Houses tied by blood as alluded to in Season 1.
Anyway, I think Tyrion hooks up with Sansa eventually, she pops out a boy which is now House Stark, House Targaryen and House Lannister. Child claims the throne since you have two of the most powerful houses in Westeros and the legitimate lineage of the royal lines in a single person. Sansa and Tyrion become the "new" Jamie and Cersei since Sansa already is getting a dark side to her.
And the cycle repeats.