"He told me the moon was an egg, Khaleesi," the Lysene girl said. "Once there were two moons in the sky, but one wandered too close to the sun and cracked from the heat. A thousand thousand dragons poured forth, and drank the fire of the sun. That is why dragons breathe flame. One day the other moon will kiss the sun too, and then it will crack and the dragons will return." -A Game of Thrones - Daenerys III
Here we have a passage of a story Doreah tells Daenerys, a tale of two moons in the sky. One wandered too close to the sun, and it cracked from the heat, resulting in dragons. And that one day, the second moon will 'kiss' the sun.
Notice G.R.R.M's play on words here. In this story, the first time one of the moons cracks, it "wandered" too close to the sun. And for the second, it is said that the second moon will "kiss" the sun. This is deliberate.
"You should look behind you, Lord Snow. The moon has kissed you and etched your shadow upon the ice twenty feet tall." Jon glanced over his shoulder. The shadow was there, just as she had said, etched in moonlight against the Wall." -A Dance with Dragons - Jon VI
Notice the same phrase of wording used here in Jon's passage. He has been "kissed" by the moon, etching his shadow along the Wall.
As I've stated in my post here, I believe that while Daenerys is the main focal point for her role as AA/TPTWP, she would be joined by others in this task. One of those people being Jon. He will be one of the three heads to join her side for the coming war against the Others. To unite the realm against the cold, and the dark.
"One day the other moon will kiss the sun too, and then it will crack and the dragons will return." -A Game of Thrones - Daenerys III
"You should look behind you, Lord Snow. The moon has kissed you and etched your shadow upon the ice twenty feet tall." -A Dance with Dragons - Jon VI
As for these two passages- let's believe they are to be taken literally. It happens for Dany and Drogo the first time around: Drogo is the sun, Dany the moon. Dany "wanders" too close to the sun, Drogo, in his funeral pyre, and thus her dragons hatch.
Now how could this relate to Jon?
I believe Jon being "kissed" by the moon, is in reference to Jon and Daenerys' eventual future romance and union together. We have evidence for this, provided from me here and here. More quotes providing into Jon being Daenerys' last romantic interest and husband:
". . . three heads has the dragon . . . the ghost chorus yammered inside her skull with never a lip moving, never a breath stirring the still blue air. . . . mother of dragons . . . child of storm . . . The whispers became a swirling song. . . . three fires must you light . . . one for life and one for death and one to love. . . Her own heart was beating in unison to the one that floated before her, blue and corrupt. . . three mounts must you ride . . . one to bed and one to dread and one to love. . . The voices were growing louder, she realized, and it seemed her heart was slowing, and even her breath. . . . three treasons will you know . . . once for blood and once for gold and once for love . . ." --A Clash of Kings - Daenerys IV
"I don't . . ." Her voice was no more than a whisper, almost as faint as theirs. What was happening to her? "I don't understand," she said, more loudly. Why was it so hard to talk here? "Help me. Show me." . . . help her . . . the whispers mocked. . . . show her . . . Then phantoms shivered through the murk, images in indigo. Viserys screamed as the molten gold ran down his cheeks and filled his mouth. A tall lord with copper skin and silver-gold hair stood beneath the banner of a fiery stallion, a burning city behind him. Rubies flew like drops of blood from the chest of a dying prince, and he sank to his knees in the water and with his last breath murmured a woman's name. . . . mother of dragons, daughter of death . . . Glowing like sunset, a red sword was raised in the hand of a blue-eyed king who cast no shadow. A cloth dragon swayed on poles amidst a cheering crowd. From a smoking tower, a great stone beast took wing, breathing shadow fire. . . . mother of dragons, slayer of lies . . . Her silver was trotting through the grass, to a darkling stream beneath a sea of stars. A corpse stood at the prow of a ship, eyes bright in his dead face, grey lips smiling sadly. A blue flower grew from a chink in a wall of ice, and filled the air with sweetness. . . . mother of dragons, bride of fire . . . mother of dragons, bride of fire . . ." -A Clash of Kings - Daenerys IV
Notice that each prophecy given to Dany in the HOTU was given to her in three, each one ending in love. When asked to be shown what it meant, Dany is given visions. Two connecting her to Jon, both ending in three, as her prophecies for love:
"Rubies flew like drops of blood from the chest of a dying prince, and he sank to his knees in the water and with his last breath murmured a woman's name" - this is clearly Rhaegar, Jon's father, dying upon the Trident. It is believed he is whispering Lyanna's name, Jon's mother.
"A blue flower grew from a chink in a wall of ice, and filled the air with sweetness" - Dany is seeing Jon at the Wall, through the metaphor of a blue flower (connecting him to Lyanna- blue winter roses). The air is full with sweetness- a metaphor to love, and sweetness- something Dany likes (also maybe this hints to Dany joining Jon at the Wall, giving him the three dragons he wishes for in another passage).
Dany will be the moon, who kisses Jon, her second sun. A reference to their love and union.
Now- how does this bring dragons into play? Who knows. The wording is a bit tricky here. The passage states that when the second moon kisses the sun, dragons will return. And yet dragons have already returned- Dany hatches her children the first time she "wanders" too close to the sun. So how do dragons come into play with her and Jon's union regarding this text?
Well, perhaps it may not be so literal. Maybe the return of dragons from Dany and Jon's union is that Jon will gain a dragon. Maybe Jon will discover ice dragons. Maybe they will find more dragon eggs at Winterfell or somewhere else. Maybe Dany's own dragons will breed and begin a second hatching of eggs, thus "returning" dragons once more with Dany and Jon's union. There are different possibilities for this.
Jon's resurrection can also lean more into him being the second sun to Dany, as he would be a wight of fire.
"Burning shafts hissed upward, trailing tongues of fire. Scarecrow brothers tumbled down, black cloaks ablaze. "Snow," an eagle cried, as foemen scuttled up the ice like spiders. Jon was armored in black ice, but his blade burned red in his fist. As the dead men reached the top of the Wall he sent them down to die again. He slew a greybeard and a beardless boy, a giant, a gaunt man with filed teeth, a girl with thick red hair. Too late he recognized Ygritte. She was gone as quick as she'd appeared." -A Dance with Dragons - Jon XII
"That night she dreamt that she was Rhaegar, riding to the Trident. But she was mounted on a dragon, not a horse. When she saw the Usurper's rebel host across the river they were armored all in ice, but she bathed them in dragonfire and they melted away like dew and turned the Trident into a torrent. Some small part of her knew that she was dreaming, but another part exulted. This is how it was meant to be. The other was a nightmare, and I have only now awakened." -A Storm of Swords - Daenerys III
"And saw her brother Rhaegar, mounted on a stallion as black as his armor. Fire glimmered red through the narrow eye slit of his helm. "The last dragon," Ser Jorah's voice whispered faintly. "The last, the last." Dany lifted his polished black visor. The face within was her own." -A Game of Thrones - Daenerys IX
Both Jon and Dany experience dreams of fighting the Others.
-Both are wearing black armor and both are wielding fire; Dany with her dragons, Jon with a sword.
And while their dreams share similarities, they also bear differences:
-Jon is battling the Others upon the Wall, whereas Dany is battling them within the Trident.
-Jon sees Ygritte and realizes too late he's killed her (a person he views with grief and regret), Dany however believes herself to be Rhaegar (a person she sees as a fierce warrior and protector).
Both are also viewed by other people as the chosen ones, AA/TPTWP:
"On Braavos, it had seemed possible that Aemon might recover. Xhondo's talk of dragons had almost seemed to restore the old man to himself. That night he ate every bite Sam put before him. "No one ever looked for a girl," he said. "It was a prince that was promised, not a princess. Rhaegar, I thought . . . the smoke was from the fire that devoured Summerhall on the day of his birth, the salt from the tears shed for those who died. He shared my belief when he was young, but later he became persuaded that it was his own son who fulfilled the prophecy, for a comet had been seen above King's Landing on the night Aegon was conceived, and Rhaegar was certain the bleeding star had to be a comet. What fools we were, who thought ourselves so wise! The error crept in from the translation. Dragons are neither male nor female, Barth saw the truth of that, but now one and now the other, as changeable as flame. The language misled us all for a thousand years. Daenerys is the one, born amidst salt and smoke. The dragons prove it." Just talking of her seemed to make him stronger. "I must go to her. I must. Would that I was even ten years younger." -A Feast for Crows - Samwell V
"Daenerys is the only hope," he concluded. "Aemon said the Citadel must send her a maester at once, to bring her home to Westeros before it is too late." -A Feast for Crows - Samwell V
"Skulls. A thousand skulls, and the bastard boy again. Jon Snow. Whenever she was asked what she saw within her fires, Melisandre would answer, "Much and more," but seeing was never as simple as those words suggested. It was an art, and like all arts it demanded mastery, discipline, study. Pain. That too. R'hllor spoke to his chosen ones through blessed fire, in a language of ash and cinder and twisting flame that only a god could truly grasp. Melisandre had practiced her art for years beyond count, and she had paid the price. There was no one, even in her order, who had her skill at seeing the secrets half-revealed and half-concealed within the sacred flames. Yet now she could not even seem to find her king. I pray for a glimpse of Azor Ahai, and R'hllor shows me only Snow. "Devan," she called, "a drink." Her throat was raw and parched." -A Dance with Dragons - Melisandre I
Jon and Daenerys both have arcs of leadership. In which both have to make hard decisions that they believe is best. Both have the qualities of strong leaders, military strategists, and unifiers. It might turn out that Jon will wind up helping Dany in the books gather the people to face against the Others and fight against the cold and the dark. Perhaps even coming into acceptance of his true parentage and relation to Dany as well.
Both Jon and Dany are also known to have cultivated into, lived with, loved, and learned the humanity of the Freefolk and the Dothraki- two factions many view as "savage" and "barbaric", and yet I believe that the Freefolk and the Dothraki will come into a big play for both Jon and Dany against the war of the Others. The culmination of their work and efforts into unifying people to work together against one common enemy.