While I like to think that the Song of Ice and Fire refers to Jon specifically, I think it also refers to the Starks and Targaryens in general.
Tada~
I am more on board with Jon Snow having a lot of relevance to The Song of Ice & Fire name. Being raised by Starks as a Stark bastard, but being Lyanna & Rhaegar's son (a hypothesis I completely buy)... And then beyond that, being the Azor Ahai prophecy reborn with the Red Woman (what more of fire), and also his connection to the Night's Watch that stands on a wall of ice.
But, really, I think that there is so much that "Ice & Fire" refers to. Winter & Summer dichotomies, dragons and others, Targaryans and Starks, so much.
Idle though, isn't there a line in one of the books or in Dunk & Egg when one character is talking to another, and she/he says, "And your song will be the song of ice and fire." ?? LIke I feel like someone says that to someone else, like maybe in Dany's visions in the House of the Undying?