A common plotline is the Asgardian princess sister coming to Earth to seek help from the Avengers. Maybe Asgard is under some sort of threat, like a new powerful enemy has emerged, and she believes the Avengers can assist. Another plotline could be her being exiled from Asgard and finding a new home and purpose with the Avengers.
One popular type of fanfiction in this category might involve the sister of Thor or Loki, who is a princess of Asgard, interacting with the Avengers. For example, she could be a powerful magic - user and help the Avengers in their battles against new threats. There are many fanfiction platforms like Archive of Our Own where you can search for such stories using relevant tags like 'Asgard princess' and 'Avengers' to find some great reads.
In fanfiction, if the Avengers go to Asgard, there could be a big clash at first. The Asgardians are proud warriors and the Avengers have their own ways. But then they might team up to face a common threat. For example, a powerful dark magic user could be targeting both Earth and Asgard. The Avengers with their diverse skills and the Asgardians with their magic and strength could form an unbeatable alliance.
One reason could be that they see Loki as a victim of circumstances. Maybe they've come to understand that his actions were often influenced by forces beyond his control, like Thanos' manipulations. So they decide to defend him in Asgard.
One possible story could be about how this Avenger, as the Princess of Asgard, uses her Asgardian powers and knowledge to help the Avengers in a battle against a powerful new enemy. For example, she might be able to call upon the Bifrost to quickly transport the team to different locations during the fight.
Probably Heimdall. He guards the Bifrost, the Rainbow Bridge that leads to Asgard. He has the ability to see all, so he'd be the first to spot the Avengers coming. And he's not one to let just anyone pass without a bit of interrogation.
You can start by looking on fanfiction websites like Archive of Our Own (AO3) or FanFiction.net. These platforms have a vast collection of Avengers fanfiction that might involve Asgard. Just use the search function on the site and type in relevant keywords like 'Asgard' and 'Avengers' to narrow down your results.
Maybe Loki is wrongly accused of a crime in Asgard. The Avengers, having had their own experiences with false accusations, step in to defend him. They believe in justice and know that Loki might not be guilty.
One popular theme is the interaction between the Asgardian characters and the Avengers. For example, Thor introducing his Asgardian culture to the Avengers team members. This can lead to interesting scenarios of cultural clash and cooperation.
It could involve her journey to Earth to join the Avengers. Maybe she was sent as an emissary from Asgard to help Earth in its time of need. This would be an interesting start as she'd have to adjust to Earth's customs and the different personalities of the Avengers.