Parents: Descended from Aerion Targaryen
Backstory and Evidence: Aerion "Brightflame" Targaryen, elder brother of King Aegon V (known as "Egg") and Maester Aemon, was known to be quite the ladies' man. By the end of his fairly short life, Aerion was also a drunk and bordering on insane. But before that, Aerion spent several years in Lys after his father exiled him there. While there, it's likely that Aerion availed himself of the city's famed pillow-houses and brothels. George R.R. Martin stated that Aerion may have fathered a few bastards in Lys, and while the timeline and ages don't match up for Varys to be one of them, he could be a descendant of that line, since he was born a slave in Lys. If Varys knows of his true heritage, he could be choosing to keep it a secret to protect himself—which also might be why he keeps his head shaven, if he actually has the silver-gold hair of a Targaryen. This could also explain why he and his friend Illyrio Mopatis have been so eager to help Daenerys regain the Iron Throne.
Verdict: Pure speculation. We don't know much about Varys' early life or parentage, that's true. But there's really nothing to connect him to Aerion Targaryen as a descendant. Even if Varys really has silver-gold hair, that doesn't mean he's a Targaryen descendant. That hair color (and purple, lilac, and indigo eyes) were commonly found among all old Valryian bloodlines. Since many of those houses fled to Lys during the Doom of Valryia, that hair and eye color combination is actually more common in Lys than anywhere else in the world. As for his motivations, I choose to believe that "show" Varys is truly motivated by his desire to have the realm be stable. In the books, his motivations are a little more complicated, but I think the end goal has been to place a puppet "fake" Targaryen on the throne—the purported Aegon Targaryen—who I believe to actually be the child of Illyrio himself.
