As you probably already know, the R+L=J theory was pretty much confirmed on the Season 6 finale of Game of Thrones.
But here's a shocker: Lyanna whispers her son's name to Ned, and while it's hard to hear, the name she whispers is definitely NOT "Jon."
Redditor sparkledavisjr has a theory as to Jon's real name, and it makes a lot of sense.
The post explains why Jaehaerys would be the probable choice for a Targaryen offspring at the time, considering recent events.
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.