Interestingly, there has been that Game of Thrones Season 6 theory running around the Internet that Lyanna Stark may have whispered a Targaryen name, Jaehaerys, into young Ned Stark’s ear just before she passed away.
If we were to presume that is true, than Ned Stark actually shortened the Targaryen name to just “Jon” and in that case would only get partial credit for naming Jon Snow.
If Robert Baratheon had found out Jon Snow was the son of (presumably) Rhaegar Targaryen all hell would likely have broken loose.
You can find out what we do know about Game Of Thrones Season 7 with our guide.