It's been established in Game of Thrones that the return of dragons spurred a resurgence in magic throughout the world (Essos and Westeros etc.), but I've been a little bothered by why the Night King chose NOW to take action after being relatively dormant for so long. So might it be that dragon's existence also triggered the Night King to make his move? In other words, did their resurgence cause HIS magic powers to increase to where he felt powerful enough to make his move now?
Not very likely
There are a few problems with this theory. Things that don't really add up in terms of chronology.
The Night King was active when there were no dragons
First of all, as @Aegon noted in the comments, the wights and White Walkers were already active in the first season, before the dragons were born in Essos.
It is a bit difficult to judge this properly, since the storylines aren't necessarily synchronised. However, based on the tidings about Dany's progress with the Dothraki in King's Landing, we know that the dragons at least were born after Ned Stark went to King's Landing. Since Ned executes Will when he is still at Winterfell, we know that the prologue took place before dragons were born. And since a white walker showed up there, it's safe to say he is already taking action at that time.
The Other slid forward on silent feet. In its hand was a longsword like none that Will had ever seen. No human metal had gone into the forging of that blade. It was alive with moonlight, translucent, a shard of crystal so thin that it seemed almost to vanish when seen edge-on. There was a faint blue shimmer to the thing, a ghost-light that played around its edges, and somehow Will knew it was sharper than any razor.
Ser Waymar met him bravely. "Dance with me then." He lifted his sword high over his head, defiant. His hands trembled from the weight of it, or perhaps from the cold. Yet in that moment, Will thought, he was a boy no longer, but a man of the Night's Watch.
The Other halted. Will saw its eyes; blue, deeper and bluer than any human eyes, a blue that burned like ice. They fixed on the longsword trembling on high, watched the moonlight running cold along the metal. For a heartbeat he dared to hope.
Prologue, a Game of Thrones
Another indication that the white walkers were active before the dragons is the massing of the wildling army by Mance Rayder. Mance took years to gather his army.
Mance had spent years assembling this vast plodding host, talking to this clan mother and that magnar, winning one village with sweet words and another with a song and a third with the edge of his sword, making peace between Harma Dogshead and the Lord o' Bones, between the Hornfoots and the Nightrunners, between the walrus men of the Frozen Shore and the cannibal clans of the great ice rivers, hammering a hundred different daggers into one great spear, aimed at the heart of the Seven Kingdoms. He had no crown nor scepter, no robes of silk and velvet, but it was plain to Jon that Mance Rayder was a king in more than name.
Jon II, a Storm of Swords
Now, this in itself isn't all that interesting, but the reason why is...
"Not at your hands." Mance studied Jon's face. "You saw the Fist of the First Men. You know what happened there. You know what we are facing."
"The Others . . ."
"They grow stronger as the days grow shorter and the nights colder. First they kill you, then they send your dead against you. The giants have not been able to stand against them, nor the Thenns, the ice river clans, the Hornfoots."
"Nor me." There was anger in that admission, and bitterness too deep for words. "Raymun Redbeard, Bael the Bard, Gendel and Gorne, the Horned Lord, they all came south to conquer, but I've come with my tail between my legs to hide behind your Wall." He touched the horn again. "If I sound the Horn of Winter, the Wall will fall. Or so the songs would have me believe. There are those among my people who want nothing more . . ."
Jon X, a Storm of Swords
Mance's conquest is for one goal and one goal only. To get the wildlings to the safe side of the wall. Why would he have started this years ago, if the White Walkers were no threat yet?
The Night King was not active when there were dragons
Up until around two hundred years ago, there were dragons in the world, but the White Walkers were already fallen in the memories of men. This in its own isn't solid proof, but it is definitely an indication, along with the previous point, that there seems to be no correlation between dragons existing and Night King attacking.
There is also no indication that the dragons affect any magic other than fire
So far, all instances of magic being stronger have been fire related. Since the overlying theme of the series is the juxtaposition of Ice and Fire, we can't just assume that the Night King would be affected as well.
The list is quite long: (Warning, potential book spoilers ahead)
- Pyromancers in King's Landing:
Tyrion was growing impatient. Ser Jacelyn Bywater was likely here by now, and Ironhand misliked waiting. "Yes, you have secret spells; how splendid. What of them?"
"They, hmmm, seem to be working better than they were." Hallyne smiled weakly. "You don't suppose there are any dragons about, do you?"
"Not unless you found one under the Dragonpit. Why?"
"Oh, pardon, I was just remembering something old Wisdom Pollitor told me once, when I was an acolyte. I'd asked him why so many of our spells seemed, well, not as effectual as the scrolls would have us believe, and he said it was because magic had begun to go out of the world the day the last dragon died."
Tyrion XI, a Clash of Kings - emphasis mine
- A Firemage in Qarth:
When the fiery ladder stood forty feet high, the mage leapt forward and began to climb it, scrambling up hand over hand as quick as a monkey. Each rung he touched dissolved behind him, leaving no more than a wisp of silver smoke. When he reached the top, the ladder was gone and so was he.
"Half a year gone, that man could scarcely wake fire from dragonglass. He had some small skill with powders and wildfire, sufficient to entrance a crowd while his cutpurses did their work. He could walk across hot coals and make burning roses bloom in the air, but he could no more aspire to climb the fiery ladder than a common fisherman could hope to catch a kraken in his nets."
Dany looked uneasily at where the ladder had stood. Even the smoke was gone now, and the crowd was breaking up, each man going about his business. In a moment more than a few would find their purses flat and empty. "And now?"
"And now his powers grow, Khaleesi. And you are the cause of it."
Daenerys III, a Clash of Kings
- Thoros of Myr, being able to revive Lord Beric Dondarrion:
"I have no magic, child. Only prayers. That first time, his lordship had a hole right through him and blood in his mouth, I knew there was no hope. So when his poor torn chest stopped moving, I gave him the good god's own kiss to send him on his way. I filled my mouth with fire and breathed the flames inside him, down his throat to lungs and heart and soul. The last kiss it is called, and many a time I saw the old priests bestow it on the Lord's servants as they died. I had given it a time or two myself, as all priests must. But never before had I felt a dead man shudder as the fire filled him, nor seen his eyes come open. It was not me who raised him, my lady. It was the Lord. R'hllor is not done with him yet. Life is warmth, and warmth is fire, and fire is God's and God's alone."
Arya VII, a Storm of Swords
- Beric Dondarrion raising Catelyn Stark:
Lady Stoneheart lowered her hood and unwound the grey wool scarf from her face. Her hair was dry and brittle, white as bone. Her brow was mottled green and grey, spotted with the brown blooms of decay. The flesh of her face clung in ragged strips from her eyes down to her jaw. Some of the rips were crusted with dried blood, but others gaped open to reveal the skull beneath.
Her face, Brienne thought. Her face was so strong and handsome, her skin so smooth and soft. "Lady Catelyn?" Tears filled her eyes. "They said . . . they said that you were dead."
"She is," said Thoros of Myr. "The Freys slashed her throat from ear to ear. When we found her by the river she was three days dead. Harwin begged me to give her the kiss of life, but it had been too long. I would not do it, so Lord Beric put his lips to hers instead, and the flame of life passed from him to her. And . . . she rose. May the Lord of Light protect us. She rose."
Brienne VII, a Feast for Crows
- An obsidian candle being lit at the citadel:
The candle was unpleasantly bright. There was something queer about it. The flame did not flicker, even when Archmaester Marwyn closed the door so hard that papers blew off a nearby table. The light did something strange to colors too. Whites were bright as fresh-fallen snow, yellow shone like gold, reds turned to flame, but the shadows were so black they looked like holes in the world. Sam found himself staring. The candle itself was three feet tall and slender as a sword, ridged and twisted, glittering black. "Is that . . . ?"
". . . obsidian," said the other man in the room, a pale, fleshy, pasty-faced young fellow with round shoulders, soft hands, close-set eyes, and food stains on his robes.
"Call it dragonglass." Archmaester Marwyn glanced at the candle for a moment. "It burns but is not consumed."
Samwell V, a Feast for Crows
All these examples are feats of magic that were suddenly made possible by the return of dragons. Notice how all of them are related to fire, and none of them related to ice.
I believe that the "Night King" was awoken when "Samuel Tarly of the Nights Watch" attacked and killed a "White Walker" with Obsidian (aka "Dragonglass") while protecting "Gilly" and her son, "Sam" in GOT S03E08. White Walkers were introduced prior to this, but this is the first act of agression towards them and certainly the first indication that they could be killed. Learning later that the Night King is connected to the White Walkers, this event would have certainly "awakened him" and created a reason to retaliate,