Bastards with no known relation to a noble house have no surname, like other smallfolk.Examples of bastard naming:The surname a bastard received appears to be connected to the location the child is raised, though this is not a consistent rule. The parents may give a bastard a different surname if they wish, e.g. Bastards with a high-born parent are given these surnames to hold them apart from their fathers' houses. It has more to do with who the bastard was born to and where they live then where they are actually born.Each of the nine constituent regions of the Seven Kingdoms have bastard surnames decreed by custom, not law. While he was physically born outside of the North, he was brought to live in the North by a Northman. Remember, Ned is claiming that Jon is his own child. There, in just a couple of months, somebody (maybe Ned, maybe not) conceived and then a woman gave birth to Jon. Ned married Catelyn in Riverrun, conceived Robb and went South to fight in Robert's rebellion.

So, why is Jon's surname Snow?There are a lot of in-universe and out-universe theoriesand even on-screen confirmation of Rhaegar+Lyannabut all of them says that Jon was born outside of the North.Update:Proof of why Jon was not born in the North Robb is a little bit older than Jon.

For example, Robert Baratheon's bastards are Edric Storm and Mya Stone. In Westeros, usually, bastards' surnames depend on the land where they were born.