During the flashback, Lyanna whispers that Robert will kill the infant if he discovers the child’s existence. We all know that Robert and House Baratheon would murder bastard children.

Considering that Rhaegar and Lyanna ran off together, it’s highly unlikely that Jon is anyone else’s child. Remember that Robert believed that Lyanna was abducted, but he probably didn’t know she had ran off with Rhaegar.

Some fans believe they heard Lyanna whisper the child’s name of “Aegon” or to Ned. While I didn’t hear it, it seems strange Rhaegar would him after his other deceased son Aegon- the one he had with Elia Martell.