If Dragonsteel is another name for Valaryian Steel the theory that Jon is Azor Ahai reborn makes even more sense, because the sword given to him by Mormont could be representative of Light Bringer.
It would be the sword that forces back the darkness (The Others).
"According to prophesy, in ancient books of Asshai from over five thousand years ago, Azor Ahai is to be reborn again as The Prince that was Promised to challenge the Others. This will occur after a long summer when an evil, cold darkness descends upon the world. It is said that The Prince that was Promised, wielding Lightbringer once again, will stand against the Others and if he fails, the world fails with him.
“ "There will come a day after a long summer when the stars bleed and the cold breath of darkness falls heavy on the world. In this dread hour a warrior shall draw from the fire a burning sword. And that sword shall be Lightbringer, the Red Sword of Heroes, and he who clasps it shall be Azor Ahai come again, and the darkness shall flee before him.""
It would be the sword that forces back the darkness (The Others).
"According to prophesy, in ancient books of Asshai from over five thousand years ago, Azor Ahai is to be reborn again as The Prince that was Promised to challenge the Others. This will occur after a long summer when an evil, cold darkness descends upon the world. It is said that The Prince that was Promised, wielding Lightbringer once again, will stand against the Others and if he fails, the world fails with him.
“ "There will come a day after a long summer when the stars bleed and the cold breath of darkness falls heavy on the world. In this dread hour a warrior shall draw from the fire a burning sword. And that sword shall be Lightbringer, the Red Sword of Heroes, and he who clasps it shall be Azor Ahai come again, and the darkness shall flee before him.""