One way could be to have Percy Jackson's powers of being a demigod interact with the superpowers of the Avengers. For example, his ability to control water could be used in a joint mission with Thor's lightning powers. Maybe they are facing a common enemy that can only be defeated by combining their unique skills.
There are so many fanfictions out there that it's hard to determine if there's a popular one with this specific plot. The Percy Jackson and Avengers fanfiction communities are quite large, but this concept goes against the general heroic portrayals in both universes. It's less likely to gain widespread popularity, but it doesn't mean there aren't some well - written ones out there that a small group of fans might really like.
Another option could be Black Widow. She's very good at adapting to new situations and using her skills in different ways. For Percy, who is coming from a very different world, she could teach him how to be stealthy and use his powers in a more covert way. She could also help him with understanding the political side of the Avengers, like dealing with government agencies and such, which Percy may not be used to in his world of demigods.
One great fanfic is 'The Demigod in the Avengers Tower'. It combines the worlds really well. The female Percy Jackson character has to navigate the new world of the Avengers while still holding onto her demigod powers. It's full of action and some really interesting character interactions.
Percy could fit in as a strategist. His experience in dealing with various gods and monsters gives him a unique perspective on handling threats. He might suggest different approaches that the Avengers haven't thought of before, like using the environment in a more creative way, similar to how he uses water in his battles as a demigod.
There are not many such fanfictions as the concept is rather dark. However, you could try searching on some major fanfiction platforms like Archive of Our Own. Use specific search terms like 'Percy Jackson abused Avengers' to narrow down the results.
In a fanfiction where Percy Jackson meets the Avengers, there would be a clash of different worlds. Percy comes from a world of Greek mythology and the Avengers are from a more science - fiction - based world. However, they could find common ground. For example, Percy's ability to withstand great physical stress could be compared to the superhuman strength of the Hulk. And his quick - thinking in battles would be on par with Iron Man's strategic mind. They could team up to face a threat that combines both magic and technology, like a villain who uses ancient spells to power high - tech weapons.
One great 'Harry Potter' fanfiction is 'The Black Family in the Modern World'. For 'Percy Jackson', 'Percy Jackson and the New Olympians' is quite interesting. And in the 'Avengers' realm, 'Avengers: Reassembled' is a popular one. These fanfictions often take the beloved elements of the original universes and add new twists and turns.
Thor might see some similarities between the Greek gods in the fanfiction and his own Norse gods. He could be laughing out loud while reading it, especially at the parts where the Greek gods act in a way that is different from what he expects. He might even start sharing stories about his own godly encounters with the other Avengers as he reads the fanfiction.
In a fanfiction, the Avengers' reactions to Percy Jackson's powers would be complex. Natasha Romanoff would immediately start thinking about how to use Percy's powers to their advantage in a mission. Bruce Banner, as the Hulk, might be a bit envious that Percy has such a natural and controlled power, while his own transformation is more of a chaotic force. Steve Rogers would be in awe of Percy's connection to an ancient power source, the Greek gods. And Clint Barton would probably make some jokes about Percy being like a human water hose, but deep down he'd be impressed. Tony Stark would want to incorporate Percy's power - controlling abilities into some new technology, or at least try to understand the science behind it.