In this post I just thought it would be interesting to run through a few events that can be more than casually linked in my opinion: The Doom of Valyria/The Targaryen Invasion of Westeros/The prophecy of the The Prince that was Promised and the Invasion of the Others/Battle for the Dawn.

Linking a Few Major Events: the Doom of the Valyria to the Prince that was Promised to the Invasion of the Others.

Note: While likely the same thing, the prophecies regarding Azor Ahai and The Prince that was Promised are different. Azor Ahai is an ancient prophecy, while the Prince that was Promised seems more recent (and specific to valyrians/targaryens).

The Doom of Valyria

We know the Targaryens fled (~116 BC) before the Doom (102 BC) due to visions by Daenys the Dreamer:

At its apex Valyria was the greatest city in the known world, the center of civilization. Within its shining walls, twoscore rival houses vied for power and glory in court and council, rising and falling in an endless, subtle, oftsavage struggle for dominance. The Targaryens were far from the most powerful of the dragonlords, and their rivals saw their flight to Dragonstone as an act of surrender, as cowardice. But Lord Aenar's maiden daughter Daenys, known forever afterward as Daenys the Dreamer, had foreseen the destruction of Valyria by fire. And when the Doom came twelve years later, the Targaryens were the only dragonlords to survive. -TWOIAF, The Reign of the Dragons: The Conquest

We also know that she had enough other dreams/visions to write a book:

"Archmaester Marwyn's Book of Lost Books." He lifted his gaze from the page to study her. "Hotho brought me a copy from Oldtown. He has a daughter he would have me wed." Lord Rodrik tapped the book with a long nail. "See here? Marwyn claims to have found three pages of Signs and Portents, visions written down by the maiden daughter of Aenar Targaryen before the Doom came to Valyria. -AFFC, The Kraken's Daughter

I think it easily can be argued that from these dreams/visions is where we get of actions by characters believing themselves (or someone close to them to be the prince that was promised). So if you are following, I am arguing that it is possible Daenys the Dreamer who prophesied the Doom, also prophesied the Prince that was Promised and the Battle for the Dawn in Signs & Portents.

I am going to list out a few characters below who seem to have at least been affected by this prophecy:

Aegon the Conqueror

Long before he chose to conquer Westeros (while the Targaryens were seemingly focused on Essos politically at the time), Aegon started plans for an invasion:

A common myth, oft heard amongst the ignorant, claims that Aegon Targaryen had never set foot upon the soil of Westeros until the day he set sail to conquer it, but this cannot be true. Years before that voyage, the Painted Table had been carved and decorated at Lord Aegon's command: a massive slab of wood, some fifty feet long, carved in the shape of Westeros and painted to show all the woods and rivers and towns and castles of the Seven Kingdoms. Plainly, Aegon's interest in Westeros long predated the events that drove him to war. As well, there are reliable reports of Aegon and his sister Visenya visiting the Citadel of Oldtown in their youth, and hawking on the Arbor as guests of Lord Redwyne. He may have visited Lannisport as well; accounts differ. -TWOIAF, The Reign of the Dragons: The Conquest

We also get this video (made while GRRM was promoting Fire & Blood) where GRRM alludes to "speculation" about Aegon having knowledge of the Battle for the Dawn/Invasion of the Others:

There is a lot of speculation that in some sense he saw what was coming 300 years later and wanted to unify the Seven Kingdoms to be better prepared for the threat that he eventually saw coming from the North, the threat that we're dealing with in ASOIAF. - George R.R. Martin on Balerion & Aegon - YouTube

I think that either a)Aegon had dragon dreams (like many other valyrians) or b) Like many Targaryens after him, he read about "the prince that was promised":

Maester Aemon's Thoughts

"You are." The woman rose in a swirl of scarlet silk, her long copper-bright hair tumbling about her shoulders. "Swords alone cannot hold this darkness back. Only the light of the Lord can do that. Make no mistake, good sers and valiant brothers, the war we've come to fight is no petty squabble over lands and honors. Ours is a war for life itself, and should we fail the world dies with us."

The officers did not know how to take that, Sam could see. Bowen Marsh and Othell Yarwyck exchanged a doubtful look, Janos Slynt was fuming, and Three-Finger Hobb looked as though he would sooner be back chopping carrots. But all of them seemed surprised to hear Maester Aemon murmur, "It is the war for the dawn you speak of, my lady. But where is the prince that was promised? -ASOS, Samwell V

While in Braavos/aboard the Cinnamon Wind, Maester Aemon drops some knowledge that not only had he been contacting Rhaegar about this (who previously believed himself to be TPTWP):

"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! -AFFC, Samwell IV

but also mentions dreams/dreamer (Daenys?) as well as a big reason why info about TPTWP isn't well know in world (Baelor likely burned most copies of Signs & Portents in addition to Barth's works):

He spoke of dreams and never named the dreamer, of a glass candle that could not be lit and eggs that would not hatch. He said the sphinx was the riddle, not the riddler, whatever that meant. He asked Sam to read for him from a book by Septon Barth, whose writings had been burned during the reign of Baelor the Blessed. Once he woke up weeping. "The dragon must have three heads," -AFFC, Samwell IV

which makes sense since Baelor was strongly against things like this:

One unfortunate aspect of King Baelor's zealotry was his insistence on burning books. Though some books might hold little that is worth knowing, and some might even hold matter that is dangerous, destroying knowledge is a painful thing. That Baelor had the Testimony of Mushroom burned is no great surprise, given its ribald and scandalous content. But Septon Barth's Unnatural History, however mistaken some of its proposals, was the work of one of the brightest minds in the Seven Kingdoms. Barth's study and alleged practice of the higher arts proved enough to win Baelor's enmity and the destruction of his work, even though Unnatural History contains much that is neither controversial nor wicked. It is only fortunate that fragments have survived, so that the lore within was not wholly lost. -TWOIAF, The Targaryen Kings: Baelor I

Aerys I, Egg & Summerhall

Like many other characters, King Aerys I was extremely interested in the "higher mysteries":

Assuming the throne in 209 AC, Daeron's second son, Aerys, had never imagined he would be king, and was singularly ill suited to sit the Iron Throne. Aerys was learned, in his way, though his interests were largely to do with dusty tomes concerned with ancient prophecy and the higher mysteries. -TWOIAF, The Targaryen Kings: Aerys I

and:

"Someday the dragons will return. My brother Daeron's dreamed of it, and King Aerys read it in a prophecy. Maybe it will be my egg that hatches. That would be splendid." -The Mystery Knight

which seemingly led to the start of Egg's interest:

The last years of Aegon's reign were consumed by a search for ancient lore about the dragon breeding of Valyria, and it was said that Aegon commissioned journeys to places as far away as Asshai-by-the-Shadow with the hopes of finding texts and knowledge that had not been preserved in Westeros. -TWOIAF, The Targaryen Kings: Aegon IV

and:

"Why did they wed if they did not love each other?"

"Your grandsire commanded it. A woods witch had told him that the prince was promised would be born of their line."

"A woods witch?" Dany was astonished. -ADWD, Daenerys IV

If interested: The Leadup to the Tragedy of Summerhall

Failed Attempts to be "reborn through salt and smoke"

While you could easily argue that this was just because they were both "Mad", but I do think you have to consider the source of where they got the idea they would be "reborn":

"Aerion the Monstrous?" Jon knew that name. "The Prince Who Thought He Was a Dragon" was one of Old Nan's more gruesome tales. His little brother Bran had loved it.

"The very one, though he named himself Aerion Brightflame. One night, in his cups, he drank a jar of wildfire, after telling his friends it would transform him into a dragon, but the gods were kind and it transformed him into a corpse. -ACOK, Jon I

and:

The traitors want my city, I heard him tell Rossart, but I'll give them naught but ashes. Let Robert be king over charred bones and cooked meat. The Targaryens never bury their dead, they burn them. Aerys meant to have the greatest funeral pyre of them all. Though if truth be told, I do not believe he truly expected to die. Like Aerion Brightfire before him, Aerys thought the fire would transform him . . . that he would rise again, reborn as a dragon, and turn all his enemies to ash. -ASOS, Jaime V

If interested: Aerion Brightflame: Connecting the Dots

Rhaegar and the Three Heads

Rhaegar (who corresponded with Maester Aemon about this) and was a survivor of Summerhall:

As a young boy, the Prince of Dragonstone was bookish to a fault. He was reading so early that men said Queen Rhaella must have swallowed some books and a candle whilst he was in her womb. Rhaegar took no interest in the play of other children. The maesters were awed by his wits, but his father's knights would jest sourly that Baelor the Blessed had been born again. Until one day Prince Rhaegar found something in his scrolls that changed him. No one knows what it might have been, only that the boy suddenly appeared early one morning in the yard as the knights were donning their steel. He walked up to Ser Willem Darry, the master-at-arms, and said, 'I will require sword and armor. It seems I must be a warrior.'" -ASOS, Daenerys I

and:

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.

and:

"Will you make a song for him?" the woman asked.

"He has a song," the man replied. "He is the prince that was promised, and his is the song of ice and fire." He looked up when he said it and his eyes met Dany's, and it seemed as if he saw her standing there beyond the door. "There must be one more," he said, though whether he was speaking to her or the woman in the bed she could not say. "The dragon has three heads." -ACOK, Daenerys IV

Marwyn

We also should note that Marwyn likely is also aware of the prophecy:

Tell me all you told our Dornish sphinx. I know much of it and more, but some small parts may have escaped my notice."

He was not a man to be refused. Sam hesitated a moment, then told his tale again as Marywn, Alleras, and the other novice listened. "Maester Aemon believed that Daenerys Targaryen was the fulfillment of a prophecy . . . her, not Stannis, nor Prince Rhaegar, nor the princeling whose head was dashed against the wall."

"Born amidst salt and smoke, beneath a bleeding star. I know the prophecy." Marwyn turned his head and spat a gob of red phlegm onto the floor. "Not that I would trust it. Gorghan of Old Ghis once wrote that a prophecy is like a treacherous woman. She takes your member in her mouth, and you moan with the pleasure of it and think, how sweet, how fine, how good this is . . . and then her teeth snap shut and your moans turn to screams. That is the nature of prophecy, said Gorghan. Prophecy will bite your prick off every time." He chewed a bit. "Still . . ." -AFFC, Samwell V

And if we go back to the original quote, I think its likely where he read about it:

Archmaester Marwyn's Book of Lost Books." He lifted his gaze from the page to study her. "Hotho brought me a copy from Oldtown. He has a daughter he would have me wed." Lord Rodrik tapped the book with a long nail. "See here? Marwyn claims to have found three pages of Signs and Portents, visions written down by the maiden daughter of Aenar Targaryen before the Doom came to Valyria. -AFFC, The Kraken's Daughter

I wanted to mention again that there is a ton of things in this post that can be attributed to Azor Ahai and not just TPTWP, I am just trying to connect some dots wrt to Daenys' visions and the actions of other Targaryens. Using GRRM's quote about "speculation" (take that how you will) about Aegon's actions, it possibly shows how serious the Targaryens were taking it.

TLDR: Daenys the Dreamer (who prophesied about the Doom) likely also was the origination of the Prince that was Promised prophecy (since it seems specific to Targaryens). This prophecy has seemingly affected House Targaryen throughout history ranging from Aegon the Conqueror to madness/misinterpretation of the prophecy.