When we talk about
born of their line I wonder if we are meant to take that literally. Looking across Jaehaerys' line of siblings rather than down the direct bloodlines instead. That would be very tricky of Martin. I also notice that Ser Duncan and Jenny of Oldstones are on that line as well and Brienne comes down from there. So I think she is destined for the Dawn Sword given the symbolism of House Tarth.
The prophecy about the prince who is promised doesn't say anything about hatching dragons but it does get mixed up with the AA legend which also says nothing about hatching dragons. Rhaelle is a sibling of Jaehaerys; it's through her line with the Baratheons that Jon is born. So maybe he is the PwiP.
Also, I was pigpiled at Westeros for this, but will repeat here.
If Dany is Rhaegar's daughter, she is still 'of the line' of Aerys and Rhaella regardless of who her mother might be, because Rhaegar is 'of the line' of sibling marriage going back two generations. Dany doesn't have to be a child of Aerys and Rhaella to fit this bill - she's still a direct descendant no matter what.
But yes, I agree - it could go crossways, although I wouldn't consider it for any Jaehaerys/Shaera's siblings (like Rhaelle) because their mother is a Blackwood, not a Targaryen, and their grandmother was a Dayne. Not enough incest in the earlier generations to fuel that. But anyone from Jaehaerys down when sibcest begins again, absolutely. They're all the same line.
Incest might be the entire problem and an outcross might be necessary for dragons to be born. What I find interesting on that horizontal line is that Duncan and Jenny of Oldstones is there as well and I think Brienne will have something to do with the Dayne sword. Someone has to be worthy of it. So then there is Aerys and his sister Rhaelle married to a Baratheon. We have two very screwed up prophecies and two potential lines for that prophecy. When Tyrion says that Jon has more of the North in him; what could be unsaid is that he has less of the dragon.
1) Dany is reborn of salt tears and smoke in MMD's tent and comet appears within a few days of her rebirth.
2) If Jon was born on the quiet Isle he is born amidst salt and smoke (although not yet reborn)
3) the wording of the prophecy changes according to who speaks it and Jon expresses it to Melisandre as born from salt and smoke. Jon is yet to be reborn and Castle Black conveniently has a smokehouse and plenty of salt blocks in their inventory. This gets back to the business of the Ironborn being reborn from salt water and what is dead can never die. I suspect Boroq will do the deed and Morna the woods witch. This could be a reprise of MMD's ritual but with certain differences.
4) Moreover AA will be reborn when the red stars bleed. That has to be associated with the brotherhood of stars and swords who carve the stars into their forehead.
5) That could very well be Euron who intends to be reborn as AA through some great magic involving holy blood sacrifice. He is the false light that Aemon fears will lead them into darkness rather than Stannis.
So I think in the end; the prophecies apply to both Dany and Jon. Everything in this book is put together with crooked stitching including the prophecies.
The Daynes may well have dragon blood. The Targaryens are the only survivors of old valyria unless the Daynes come from a line of dragon blood. Dany's wake the dragon dream suggests just that:
A Game of Thrones - Daenerys IX
She was walking down a long hall beneath high stone arches. She could not look behind her, must not look behind her. There was a door ahead of her, tiny with distance, but even from afar, she saw that it was painted red. She walked faster, and her bare feet left bloody footprints on the stone.
"You don't want to wake the dragon, do you?"
She saw sunlight on the Dothraki sea, the living plain, rich with the smells of earth and death. Wind stirred the grasses, and they rippled like water. Drogo held her in strong arms, and his hand stroked her sex and opened her and woke that sweet wetness that was his alone, and the stars smiled down on them, stars in a daylight sky. "Home," she whispered as he entered her and filled her with his seed, but suddenly the stars were gone, and across the blue sky swept the great wings, and the world took flame.
"… don't want to wake the dragon, do you?"
Ser Jorah's face was drawn and sorrowful. "Rhaegar was the last dragon," he told her. He warmed translucent hands over a glowing brazier where stone eggs smouldered red as coals. One moment he was there and the next he was fading, his flesh colorless, less substantial than the wind. "The last dragon," he whispered, thin as a wisp, and was gone. She felt the dark behind her, and the red door seemed farther away than ever.
"… don't want to wake the dragon, do you?"
Viserys stood before her, screaming. "The dragon does not beg, slut. You do not command the dragon. I am the dragon, and I will be crowned." The molten gold trickled down his face like wax, burning deep channels in his flesh. "I am the dragon and I will be crowned!" he shrieked, and his fingers snapped like snakes, biting at her nipples, pinching, twisting, even as his eyes burst and ran like jelly down seared and blackened cheeks.
"… don't want to wake the dragon …"
The red door was so far ahead of her, and she could feel the icy breath behind, sweeping up on her. If it caught her she would die a death that was more than death, howling forever alone in the darkness. She began to run.
"… don't want to wake the dragon …"
She could feel the heat inside her, a terrible burning in her womb. Her son was tall and proud, with Drogo's copper skin and her own silver-gold hair, violet eyes shaped like almonds. And he smiled for her and began to lift his hand toward hers, but when he opened his mouth the fire poured out. She saw his heart burning through his chest, and in an instant he was gone, consumed like a moth by a candle, turned to ash. She wept for her child, the promise of a sweet mouth on her breast, but her tears turned to steam as they touched her skin.
"… want to wake the dragon …"
Ghosts lined the hallway, dressed in the faded raiment of kings. In their hands were swords of pale fire. They had hair of silver and hair of gold and hair of platinum white, and their eyes were opal and amethyst, tourmaline and jade. "Faster," they cried, "faster, faster." She raced, her feet melting the stone wherever they touched. "Faster!" the ghosts cried as one, and she screamed and threw herself forward. A great knife of pain ripped down her back, and she felt her skin tear open and smelled the stench of burning blood and saw the shadow of wings. And Daenerys Targaryen flew.
"… wake the dragon …"
The door loomed before her, the red door, so close, so close, the hall was a blur around her, the cold receding behind. And now the stone was gone and she flew across the Dothraki sea, high and higher, the green rippling beneath, and all that lived and breathed fled in terror from the shadow of her wings. She could smell home, she could see it, there, just beyond that door, green fields and great stone houses and arms to keep her warm, there. She threw open the door.
"… the dragon …"
A Game of Thrones - Daenerys III
Day followed day, and night followed night, until Dany knew she could not endure a moment longer. She would kill herself rather than go on, she decided one night …
Yet when she slept that night, she dreamt the dragon dream again. Viserys was not in it this time. There was only her and the dragon. Its scales were black as night, wet and slick with blood. Her blood, Dany sensed. Its eyes were pools of molten magma, and when it opened its mouth, the flame came roaring out in a hot jet. She could hear it singing to her. She opened her arms to the fire, embraced it, let it swallow her whole, let it cleanse her and temper her and scour her clean. She could feel her flesh sear and blacken and slough away, could feel her blood boil and turn to steam, and yet there was no pain. She felt strong and new and fierce.
And the next day, strangely, she did not seem to hurt quite so much. It was as if the gods had heard her and taken pity. Even her handmaids noticed the change. "Khaleesi," Jhiqui said, "what is wrong? Are you sick?"