Have you heard the one that Tyrion is targaryen as well?
Aerys, the mad king, seduced or outright raped Joanne Lannister. In attempting to force a miscarriage, Tywin gave her some kind of potion/chemical that only weakened her and stunted Tyrion's fetal development, in her weakened state she would later die giving birth to him.
Multiple parts of the theory -
Tywins resentment and outright claims of "you are no son of mine" to Tyrion bolster the notion Joanne was seduced and unfaithful, Tywin views it as a disgrace and his attempt to abort Tyrion only resulted in Joannes death, leaving Tyrion as the constant reminder of his wife's indiscretions and his killing her attempting to abort him.
When Robert was crushing the Targaryen empire and his ethnic cleansing was progressing well, Tywin marched on Kings Landing where he would extract his revenge for Joanne, claiming he was there to help protect the city but intended on sacking it once the gates were open.
Jamie was still in service of Aerys, knew what his father was attempting and advised Aerys to not open the gates to his father's army. Whatever transpired in the great Hall, Jamie planned on honoring his vow to protect the king but ended up killing him. Maybe the King mocked him, maybe Tywin got word to him, but he found out about his mother and killed Ares. Jamie maintained a soft heart in regards to Tyrion as he was not to blame, just a victim of consequence.
Now let's talk about the Targaryens.
The Targaryen banner is a three headed dragon, red in color, with the motto "Fire and Blood. Dragons must be born of blood, there are three dragons and by the theory, there are three remaining Targaryens.
Jon, Tyrion, and Daenerys were all born of blood, their mothers died giving birth to them.
How this could play into the series end, I have not a fucking clue.