In the Tower of Joy sequence from the finale of Game of Thrones season 6, we only hear some parts of what Lyanna whispered to young Ned. We hear her starting the sentence with ‘His name is’ (see this video, skip to 3:40) and then we hear ‘if Robert found out, he will kill him, you know he will. You have to protect him. Promise me Ned. Promise me’.
Now, check out this fan theory by Redditor sparkledavisjr, who tries to complete Lyanna’s first sentence. She has rewatched the scene countless times and thinks that Jon Snow’s given name at birth by his mother was Jaehaerys:
“I have rewatched the new TOJ scene about a dozen fifty times at least now, with captions and without. Lyanna starts by whispering “His name is”, and then what follows appears to be a three syllable word that appears to start with a J. I’m not a great lip-reader, but I’m fairly certain what she says is Jaehaerys. The name is not included in the captions, probably on purpose. Watch the scene starting here: https://youtu.be/Y3IwIB7yfO8?t=3m40s.
This makes a lot of sense as a name (instead of Jon), as both previous Jaehaerys to rule were considered good Kings.
Jaehaerys I was the 4th Targaryen king to sit the Iron Throne. He was known as “the Wise”, “The Conciliator”, and “the Old King”. His long rule was prosperous and he was aided by his sister-wife Alysanne, who convinced Jaehaerys to expand the Night’s Watch and granted them the land now known as the New Gift.
Jaehaerys II was Jon’s great-grandfather. His reign was short due to his ill-health (only 3 years), but still a good one. Jaehaerys restored stability to the realm, ended the Blackfyre threat during the war of the Ninepenny kings, and improved relations with the major houses who were unhappy during his father’s rule (Aegon V – Egg, the brother of Maester Aemon).
If Lyanna was in love with Rhaegar (as seems likely), it makes sense for their son to have a Targaryen traditional name like his siblings Aegon and Rhaenys. Jaehaerys would be a good pick as the name isn’t associated with any real negative connotations, like Aerys, Maegor, etc. Not only is Jaehaerys the great-grandfather of Jon, he was the King who received the ORIGINAL PROPHECY about the prince who was promised from the woods witch with Jenny of Oldstones.
This prophecy is what caused Jaehaerys to force his children Aerys and Rhaella to marry, because it was prophesied that the PTWP would be born of their line. Rhaegar was aware of this prophecy, and while he originally thought that he was the PTWP, he realized later he was not and that it would be one of his children. That is why he took/seduced Lyanna in the first place. His wife Elia could not have any more children, and he knew there must be a third child (The dragon must have three heads).
It makes sense that Ned knew that he obviously couldn’t keep this name if he wanted to keep the boy’s true parentage a secret. As a result, he re-named him Jon. There are two people he could possibly be named after in the Stark tree– Jon Stark was a King in the North who drove back raiders at White Harbor and built the “Wolf’s Den” on the White Knife. Jonnel Stark was a Lord of Winterfell after it became part of the Seven Kingdoms. He was married twice– once to Robyn Ryswell and once to his niece Sansa Stark. Neither marriage had issue, so he was followed as Lord of Winterfell by his brother Barth Blacksword.”