How about being a foreign diplomat from Essos? You come to Westeros with the intention of forging trade deals but end up getting involved in the power play for the throne. You could introduce new customs and ideas from Essos, and also have to deal with the xenophobia that exists in Westeros while trying to achieve your own goals.
One idea could be a young child from our world being inserted into the story during the events at King's Landing. They could befriend Arya Stark and help her in her quests, perhaps by using their knowledge of modern things in a creative way, like making simple traps with modern concepts. Another plot could involve the kid getting caught up in the politics of the North, trying to convince the lords to unite against the White Walkers earlier. Or they might be found by Daenerys and become a sort of mascot for her army, bringing a new perspective to her claim for the Iron Throne.
One idea could be that your self - inserted character is a healer with a mysterious past. They end up in the middle of the war between the Starks and the Lannisters and start to form relationships with key characters from both sides, creating a harem of sorts. Another plot could be that your character is a time - traveler who tries to change the events of 'Game of Thrones' and in the process, attracts the attention of different characters who fall for them.
One idea could be to insert yourself as a commoner in King's Landing. You could be a street - smart character who witnesses all the political intrigue up close. Maybe you work in a tavern and overhear the schemes of the nobles. Another idea is to be a foreigner from Essos who comes to Westeros with unique skills, like being an expert in a form of magic unknown in Westeros. You could then get involved in the power struggles while trying to keep your own identity and goals secret.
One plot idea could be that your self - inserted character is a Lannister bastard who is trying to prove their worth to the family. Another is that your character discovers a long - lost Lannister treasure and has to fight off other family members who want it for themselves. Or perhaps your character is a Lannister who has a secret alliance with a rival house and is trying to bring down the main Lannister power from within.
You could insert yourself as a human knight. Picture this: you are part of the Stormwind army, and you get involved in the fight against the orcs. During a battle, you find a mysterious artifact that gives you special abilities. This leads you on a quest to find out more about the artifact's origin and how it can be used to save humanity from various threats like the Scourge or the demons.
Your self - insert could be a transfer student from another kingdom with a completely different fighting style. This would lead to interesting interactions with the main characters as they learn from each other. They might introduce new weapons or combat techniques that the others haven't seen before.
A possible plot is that the self - inserted character has to deal with the Omnitrix malfunctioning. This could lead to unexpected and sometimes comical transformations. They might transform into a really weak alien in the middle of a battle or into an alien that is completely inappropriate for the situation, like a tiny alien when they need to be strong. As they try to fix the Omnitrix, they learn more about its inner workings.
The plot might involve the self - inserted character being found by a group of wildlings after the plane crash. Since they come from the modern world, they might be seen as some sort of strange being. But they could start to learn the ways of the wildlings and use their unique skills, like being able to read maps better or having a better understanding of the stars for navigation, to help the wildlings. And in return, the wildlings could protect them from other threats in the Game of Thrones world.
In 'A Game of Thrones self - insert fanfiction', themes like adapting to a new culture are common. Since Westeros has its own unique set of rules, religions, and social norms, the self - insert character has to learn and adapt. Betrayal is another theme. Given the cut - throat nature of the politics in the series, it's likely that your character will experience or be involved in some form of betrayal. And the theme of prophecy can also be used. Maybe your character is somehow related to an old prophecy in Westeros and has to fulfill it or prevent it from coming true.
One idea could be exploring what would happen if a minor character like Gendry had been secretly trained in magic. He could use his newfound powers to unite the smallfolk against the warring nobles. Another plot could revolve around a female knight from the North who disguises herself as a man to fight in the wars, and along the way, she discovers secrets about the White Walkers that no one else knows. Or perhaps a fanfiction where Daenerys and Sansa form an alliance much earlier and together they reshape the power dynamics of Westeros.