Obviously you have the R+L=J theory within the A Song of Ice and Fire fandom. basically it states that Jon Snow is actually the son of Rhaegar Targaryen and Lyanna Stark. I personally think this is the case but it still has yet to be addressed.
Of course, there are also the theories that make you want puke, namely the idea that Tyrion Lannister is secretly the rape baby of Aerys Targaryen and Tywin wife Joanna and therefore he is a secret Targ. This just seems to be fans wanting their favorite character to be even more special and I think it would turn Tyrion into a total Marty Stu (or at least more so that he already is).