Here's Why Neither Daemon Nor Aemond Can Be Night King
No matter how much you want it.
For some unknown reason, House of the Dragon has sparked a wave of fan theories that suggest that the Night King from the Game of Thrones might actually be one of the Targaryens that are shown in the prequel.
The most popular candidates for being Night King are Daemon Targaryen and Aemond Targaryen — probably because they do share a certain resemblance with the Night King from Game of Thrones. However, this is where the similarities end (except that yes, Aemond does have that sapphire eye that so looks like the blue eyes of the white walkers...)
No matter how into you are in such fan theories, it is highly unlikely they will ever be confirmed. First of all — because the Night King actually came into existence long before the said Targaryens were even born.
The night kind was indeed a mortal man once (another argument that those believing Daemon or Aemond could be him use frequently), but he existed since the age of the First Men, which is thousands of years before the events of House of the Dragon.
However, those who don't care much about the book canon or simply don't know it have already established their head canon, and it seems nothing can persuade them.
Fans who believe that Daemon is the Night King point at the similarities in their behaviors, and they also claim that when Daenerys tried to roast the Night King in Game of Thrones, he was immune to fire — just like Targaryens are.
"Daemon’s clothing is similar to the night king. The collar with the shoulders. Textures are similar. Plus the Night King rode the dragon like he’d done it before. I agree and thought immediately Daemon is the Night King." – u/FalconOk2795
Those who believe that Aemond is the Night King also seem to be overly into antagonists with at least one blue eye. Still, no matter what path House of the Dragon picks, the book canon rules out Daemon or Aemond being the Night King.
The second season of House of the Dragon is due to start filming in early 2023, with the premiere date to be scheduled by HBO Max later. As of now, the prequel seemed to have largely followed the books, so there are no indicators that could imply that the showrunners would want to ditch the canon and make Daemon or Aemond Night King.