From this article:
The Targaryens, for those of you who lose track of all of the surnames in the series, are the platinum-haired royal family who ruled the land for centuries prior to the events of the show/books. Daenerys' entire claim to the throne is based on the idea that she's the last surviving member of that family. She intends to press this claim using her inherent genetic ability to befriend dragons, which is how the Targaryens came to power in the first place.
So, anyone who turns out to be a Targaryen in this universe automatically is a big deal, because they can make a plausible claim to the throne. According to Internet theorists, the reason Tyrion is everyone's favorite Lannister (aside from the stone-fisted awesomeness of Peter Dinklage) might be because he isn't actually a Lannister at all -- he's half Targaryen.
First, there's the fact that Tywin, Tyrion's official father, hates Tyrion with every fiber of his being and loathes calling him "son." He even says to Tyrion, "Men's laws give you the right to bear my name and display my colors since I cannot prove that you are not mine."
Now, you could just chalk that up to another gilded gauntlet full of cold-blooded spite Tywin is so fond of hurling around, but the fact that he said "I cannot prove that you are not mine" seems to indicate that he really doesn't think Tyrion is his child.
See, in the books, Tyrion is described as having hair so blond that it's nearly white (all the Targaryens have white hair), as well as one green eye and one dark eye that's almost black (it could conceivably be a deep purple, which is the color of the Targaryens' eyes). Add to that the fact that King Aerys Targaryen, whom Tywin worked for, totally wanted to bang Tywin's wife, and the evidence begins to stack up to right around waist height -- the former king is Tyrion's real dad.
Some fans even extend the theory to Tywin's less-shame-inducing offspring, Jaime and Cersei. If you think about it, they certainly display some undeniably Targaryen characteristics: They carry on the age-old Targaryen tradition of hyper incest, they're both absurdly narcissistic, and Cersei is obsessed with fire. Jaime even compares her to King Aerys after she burns down the Tower of the Hand, and Jaime would know, because he personally stabbed King Aerys to death.
The real smoking crossbow, though, is that on Tywin and Joanna's wedding night, Aerys said that it was a "great pity that the lord's right to the first night had been abolished" before taking "liberties ... during the bedding." Exactly what sort of "liberties" are unclear, but it's entirely possible that not a damn one of Tywin's children are actually his.
Is there any evidence that any (or all) of the Lannister siblings are really Targaryens?