Could Ned's promise to Lyanna tie at all into "The prince who was promised"?
Yes. If there is an Azor Ahai/Prince that was Promised it's Jon. A lot of people think it's Dany, which is pretty credible, but I'd bet on Jon. The aspects of the theory are as such:
Prince: Jon is the son of Rhaegar ergo a prince (though Aemon says gender doesn't matter)
Blood of the Dragon: Jon is half Targ
Born amidst salt and smoke: When Jon was totally not killed, the salt comes from the tears of Bowen Marsh and the smoke relates to how Jon describes the cold "smoke" coming from his wounds.
Bleeding Star: That guy Wun Wun kills has a star symbol on his arm, the arm that the giant ripped off.
Other things: Mel sees Jon Snow in her visions searching for AA, Lightbringer could be the Night's Watch etc.
You can't really count on prophecies being 100% true, but so far in the story every prophecy has had some truths to it. Hell even Patchface gets stuff right and he's crazy.