Perhaps Harry is a Stark and the cave is a place of prophecy. He enters the cave and sees visions of the future of Westeros, including the downfall of his family if certain actions are not taken. He then uses his knowledge from his previous life (if we assume some sort of reincarnation or transfer from his Harry Potter world) and his Stark upbringing to change the course of events. He might form alliances with unlikely characters, train in new forms of combat or magic found in the cave, and ultimately strive to put a Stark on the Iron Throne, all while trying to keep his family alive and the realm in balance.
Yes, there are quite a few out there. You can start by looking on fanfiction websites like Archive of Our Own or FanFiction.net. Just search for the specific tags 'Harry', 'Stark', 'Cave' and 'Game of Thrones' to find relevant stories.
It could be that Arya is tape - gagged during a mission gone wrong. She was infiltrating an enemy stronghold but was discovered. Now, she has to figure out how to free herself and complete her mission while being silenced. This would show her determination and survival skills even in a difficult situation.
One possible plot could be that Robb Stark manages to avoid the Red Wedding trap. Maybe he receives a tip - off from an unexpected source, like a disgruntled Frey servant. With this knowledge, he could turn the tables on his enemies. He might use his military strategy skills to ambush the Freys instead, and this could lead to a major shift in the power dynamics in the North, with Robb emerging as an even stronger leader.
The male OC Stark heir could be thrust into the role of leadership much earlier than expected due to unforeseen circumstances, like the sudden death of his father. He then has to deal with the challenges of ruling Winterfell, such as food shortages, disputes among the vassals, and the ever - present threat of the White Walkers. He uses his unique skills and the support of his loyal followers to overcome these obstacles and restore the Stark name to its former glory.
One possible plot could be Arya Stark rebuilding the Stark legacy all by herself. She might use her skills as a Faceless Man to secretly eliminate threats to Winterfell and the North. For example, she could target those who had a part in the downfall of her family during the chaos of the Game of Thrones. With her unique abilities, she would be able to move in the shadows and slowly but surely regain the power and respect that the Stark name once held.
Well, in this fanfiction, they might be lured into the cave by a mysterious figure. Once inside, they find that the cave has some sort of mind - altering properties. Clint starts to see visions of his past failures, and Tony is haunted by the idea that his technology could be used for evil. They have to overcome these mental challenges and find a way out of the cave together. They rely on their friendship and mutual respect to get through it all. It could also show how they grow as characters during this ordeal.
One emotional moment could be when Arya sees Sansa again. After all they've been through, the estrangement, the different paths they took. When they meet, all the past grudges are forgotten, and they just hold each other, crying. It's a moment of pure sisterly love and relief that they are both alive and together again.
One cool plot could be that Tony Stark is transported to Westeros right before the onset of the Long Night. He teams up with Bran Stark, using his technology to help Bran better understand the visions he has of the past and future. Stark also uses his suit to fight the White Walkers, creating a new type of weapon against the ice - based enemies. His presence also causes a stir among the various houses, who either want to use his power for their own gain or see him as a threat.
The plot could involve Arya being kidnapped by a group of bandits. They gag her to keep her quiet as they plan to ransom her. But Arya uses her training to manipulate the situation. She might try to communicate with other captives or find a way to get the gag off and fight back.
An interesting plot could be that the Stark family discovers a portal to the Naruto universe. Sansa Stark, being the politically astute character she is, tries to learn about the power structures in the ninja world. Meanwhile, Naruto and his friends are intrigued by the Starks' noble demeanor. They form an alliance to fight a common enemy that somehow threatens both their worlds, like a powerful dark force that has the ability to manipulate time and space.