[@agentmanatee] Yes, Lyvia is perhaps one of the many bastards sired by Ser Gregor Clegane, the Mountain that Rides. Lyvia in particular had been sired between Ser Gregor and a brewer's daughter named Layna (http://awoiaf.westeros.org/index.php/Layna).  

[@MrDidact] Whilst normally a bastard would only have the more base born name of Hill or Rivers, who would be around to disprove the claim of Lyvia? Ser Gregor is dead, and no doubt all of his men are dead as well, either through Arya killing them off due to her list, or when defending Cersei during the war of reconquest, directly related to combat deaths, executions for their war crimes, or if they somehow survived, they'd be a the wall, so far away.

Once I finish writing their character biographies, it will make sense. For now, the aura of mystery hangs like a thick fog.