Bastards are not named for where they are born but where they are raised.
Remember, Ned is claiming that Jon is his own child. While he was physically born outside of the North, he was brought to live in the North by a Northman. It has more to do with who the bastard was born to and where they live then where they are actually born.
Explanation of Bastard Naming:
Each of the nine constituent regions of the Seven Kingdoms have bastard surnames decreed by custom, not law. Bastards with a high-born parent are given these surnames to hold them apart from their fathers' houses. The parents may give a bastard a different surname if they wish, e.g. Tyrion Tanner. 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. Because of this, bastards who are half-siblings might have different surnames. For example, King Robert I Baratheon's eldest bastard, born in the Vale, is called Mya Stone, while his bastard from the Stormlands (fathered on a noblewoman from the Reach) is called Edric Storm. However, whilst Aegor Rivers was raised at Stone Hedge in the Riverlands, his half-brother Brynden, was also called Rivers, despite having lived at least the first few years of his life with his mother at King's Landing in the Crownlands.
So it would have depended on which parent took custody of Jon.
If it had been Rhaegar, Jon would have likely been a Waters since Rhaegar would have taken him back to the Crownlands.
If it had been Lyanna, Jon would have likely still been a Snow since he would have been brought back to the North.