I'm pretty firmly on the "Jon Snow doesn't have a "real" Targaryen name, it's actually pretty important that he is Jon Snow and if Lyanna gave him a name, it'd be more likely to be a Northern name" train. However, if he did have a secret Targaryen name, the most correct answer is Aemon because that makes two Aemon Targaryens sworn to the Watch, having given up everything in actual service to the realm. The second most correct answer is actually a Velaryon name but you know what, it would be fucking hilarious if there was a second not-so-secret Targaryen bastard named Jacaerys who's nevertheless the most employed Targaryen of his generation. Also, Jacaerys is an objectively better letter J-starting namesake compared to the low, low bar set by Jaehaerys I (trades title of Westeros' worst father back and forth with Tywin Lannister) and Jaehaerys II (reinvented sibling incest after several generations of doing marginally better).