Well, a plot for such a fanfiction could start with the kid being magically transported to Westeros just as the War of the Five Kings is about to begin. The kid could end up in the care of the Lannisters. Since they are shrewd, they might use the kid as a symbol of their 'benevolence' towards the common folk. But the kid, being from our world, would see through their schemes and try to escape, all while getting involved in various intrigues like the trial of Tyrion. They could also have some influence on the relationship between Jaime and Cersei, maybe by making Jaime question his loyalty to his sister more quickly.
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.
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 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.
One idea could be an engineer SI who uses their knowledge to improve the defenses of a particular house in Westeros. For example, they could design more efficient siege weapons or fortifications that are resistant to dragon fire. This would add a new dimension to the power struggles in the Seven Kingdoms.
The character can start off as naive and confused, like any kid would be in a strange new world. As they experience the harshness of Westeros, they gradually become more street - smart. For example, they might learn how to barter and negotiate from the people in the markets. They could also develop morally. At first, they might be hesitant to use violence, but as they face threats, they learn when it's necessary. Their relationships also shape them. If they befriend a noble, they might learn about courtly manners and power plays, while befriending a commoner would teach them about the daily struggles.
One interesting 'Game of Thrones' SI fanfiction could be the ones where the self - inserted character tries to change the fate of their favorite characters. For example, preventing Ned Stark's death. Another might be about a modern - day person using their knowledge to introduce new technologies in the 'Game of Thrones' world, like advanced medicine. And there are also those where the SI character gets involved in the power struggles, like trying to claim a throne for themselves in a unique way.
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.