A fan by the username 'Sparkledavisjr' tried to read the lips of Jon Snow's dying mother to find out his real name. “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,” explains the fan.
He continues: 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 three 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.
It does make sense that Ned renamed the boy Jon, to keep him safe. Ned Stark probably chose the name Jon after Jon Arryn, Ex-Hand of the King Robert Baratheon, who was to Ned a father figure and good friend.