Is this what Jon's name should've been?

Question

In "The Dragon and the Wolf" (Game of Thrones, S07E07), Bran tells Sam that Jon is actually Rhaegar Targaryen and Lyanna Stark's son, and that his name should've been Jon Sand. Of course, Sam corrects him by saying Jon isn't a bastard at all.

But is "Sand" really what Jon's last name would've been if Rhaegar and Lyanna wouldn't have been married? On what basis is a bastard named (last name)?

Answer 1

IF Jon had been a bastard and acknowledged by his father he would have taken / been given the common name for bastards in the region where he would have been born.

Bastards only use the special surnames if they have been openly acknowledged by their noble-born parent. In such cases, their noble parent will usually try to make sure that they are well cared for, or send money for their support, but it is extremely unusual for a noble to raise their bastard child in their own household.

^Wikia

Lyanna gave birth in Dorne and the bastard's names go by region of birth in GOT.

• Flowers: The Reach
• Hill: The Westerlands
• Pyke: Iron Islands
• Rivers: The Riverlands
• Sand: Dorne
• Snow: The North
• Stone: The Vale of Arryn
• Storm: The Stormlands
• Waters: The Crownlands

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.

^Source

Answer 2

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.

Answer 3

In the books it depends on the region of the family, but Targaryens had a special name for their bastards, that being Blackfyre, so i suppose Jon would be a Blackfyre since Rhaegar is his father.