The House of the Dragon trailer may have a connection to Arya killing the Night King in Game of Thrones. The first of many planned Game of Thrones spinoffs, House of the Dragon takes place around 200 years before the main show, and will be centered around House Targaryen. Given the timeline, then it's going to be a rather different show focused on a whole new set of characters, but some elements will be familiar to viewers.
As shown in the House of the Dragon trailer, there'll still be recognizable themes and iconography. In particular, the Iron Throne will be crucial to the prequel's story, as the battle of succession leads to the Targaryen Civil War known as the Dance of the Dragons. That name, somewhat tellingly, reveals another key way this will be like Game of Thrones, as audiences can once more expect a lot dragon action, on an even greater scale than what was delivered via Daenerys Targaryen.
Those are the clearer links, but House of the Dragon's trailer hints at another connection to Game of Thrones, thanks to the presence of what appears to be a Valyrian steel weapon. In the footage, a panicked Alicent Hightower (Olivia Cooke) bursts into what seems like a council meeting holding a dagger. This doesn't seem to be just any old blade, though; the design of it, including the ruby in the center of its hilt, means it looks a lot like the catspaw dagger from Game of Thrones. The Valyrian steel weapon first featured in season 1, where it was used in the assassination attempt on Bran Stark, and it made it all the way to season 8, as Arya Stark killed the Night King with it.
That begs the question of what the Valyrian steel dagger from Game of Thrones would be doing in House of the Dragon, but its presence would actually make sense. While the dagger had belonged to Littlefinger, its origins are clearly much grander. That it's made of Valyrian steel, along with a dragonbone hilt, points to Valyria and most likely the Targaryens themselves, meaning it was likely handed down through the generations, and then perhaps ended up in Littlefinger's possession after the fall of the Targaryen dynasty with the death of the Mad King, Aerys II Targaryen. Of course, after that it was passed to Arya, and ultimately caused the Night King's death. In Game of Thrones season 7, Samwell Tarly comes across an image of the dagger in a book at the Citadel, which reads:
"When Aegon the conqueror forged his Seven Kingdoms, he and his descendants would often decorate their blades with dragonglass feeling a kinship with the stone. The royal fashion for dragonglass ornamentation soon spread throughout the Seven Kingdoms to those wealthy enough to afford it. Hilts and pommels were and are the most common decoration for dragonglass if too brittle to make a useful crossguard. Indeed, its very brittleness is what relegate it to the great houses and the most successful merchants."
That supports the notion that the Valyrian steel dagger came directly from House Targaryen, and perhaps even Aegon himself. Of course, it is possible that the weapon in House of the Dragon is simply very similar to the one from Game of Thrones, but given the shared DNA of the shows and how seamlessly it fits into the story, it would fit for them to be one and the same. That's the kind of fun reference House of the Dragon can do, where it serves a purpose to the story and isn't too overt in its winking at the viewer. As for just how it fits into the prequel's narrative, the scene in question is perhaps the one where Alicent's son, Aemond, has lost an eye in a fight with the children of her rival, Rhaenyra Targaryen (Emma D'Arcy), further widening the division and stoking the fires that will eventually erupt into the Dance of the Dragons.