Waters for King’s Landing and Dragonstone, Snow for the North, Pyke for the Iron Islands, Rivers for the Riverlands, Stone for the Vale, Hill for the Westerlands, Flowers for the Reach, Storm for the Stormlands and Sand for Dorne. Some believe that the bastard name for Targaryen bastards is Blackfyre, however Blackfyre instead was a name which Aegon IV’s bastard son Daemon was allowed to take upon being knighted; prior to this, he was Daemon Waters. This is supported by the fact that his half-brothers Aegor and Brynden used the Rivers name, because their mothers were of the riverlands, and Princss Elaena Targaryen’s bastard children by Lord Oakenfist had the last name of Waters.
It’s possible that this picture is being complicated with A Dance with Dragons, as Lollys Stokeworth’s son is noted as “Tyrion Tanner”. There’s a possibility that this is another bastard name, perhaps unique to King’s Landing. However, the fact that Lollys conceived the child after being raped behind a tanner’s shop may make it likelier that this is a kind of mocking byname that the child has already been bestowed.