She LITERALLY trained for many seasons to be an elite assassin in the Faceless Men that LITERALLY worshiped the Many Faced God aka The Lord of Light aka The God of Death.
I already answered if Arya qualifies for being Azor Ahai
Let's argue for a moment on potential candidates to kill the Night King: Jon, Daenerys, Jaime (being Kingslayer), Brienn (together with Jaime tie-ing up Nissa Nissa if you want to take prophecies. I don't care about prophecies in GoT).
After the battle began, it was pretty clear that Jaime and Brienn were only soldiers in the battle because of the strategy used by the Night King of letting the Wights loose on the Army of the Living and keeping himself and other White Walkers far back.
So, only Jon and Daenerys, on dragons are left. Daenerys tries Dracarys on Night King and it doesn't work. She can't do anything more so she's out and we are left with Jon.
When the Night King crash lands and is making his way up to Bran, Jon seeks him out by running behind him, but the Night King senses this and just resurrects some more wights for Jon to fight through and goes on his mission.
The Night King is very clear in his strategy. Use as many Wight soldiers in this fight for as long as you can and keep the White Walkers at a fair distance from the actual fight.
We see in earlier seasons that White Walkers are not skilled fighters. They never had to be. Their weapons were indestructible and steel blades or fire couldn't harm them. But Jon easily killed a White Walker at Hardhome with Valyrian Steel and Night King saw it. Another White Walker was ambushed and killed by Jon and co beyond the Wall. Another was killed by Meera through a spear from long distance.
So, Night King knows that White Walkers are now vulnerable. He has a clear strategy of using only Wights. He has seen Daenerys beyond the Wall and knows that her dragonfire would not be effective.
So, to sum up, he avoids one-on-one fight with Jon as well as his other White Walkers because he knows Jon will kill them and him in single combat.
So the attack on the Night King had to be a sneak attack. It had to be something he could not predict or prepare for. So, Arya was the only prime candidate to kill him.
Theon, instead of charging, could have just aimed the spear at the Night King but he was a fair distance away and could avoid it.
The plan of the Army of the Living was to kill the Night King through dragon fire. If that fails, it was up to Jon to kill Night King through single combat. The Night King could anticipate these plans but not anticipate being killed by a sneaky assassin.
So, Arya killing the Night King makes perfect sense because she trained to be an elite skilled assassin who could sneak up on anyone without making a sound.
Arya is not the Deus Ex Machina or Mary Sue as many fans choose to believe. She was the only skilled person among the living to kill the Night King