I'm not sure specifically if there is one with that exact title, but considering how fanfic writers love to mix and match different fandoms, it's very likely. Writers often take the characteristics of one character, like Luna from Harry Potter, and impose them on another character, in this case Harry Potter, and then place them in a new universe like the Avengers. It could be a really fun read if it exists.
Yes, there might be. The world of fanfiction is vast and diverse. With the popularity of both Harry Potter and Avengers, it's not uncommon for fans to create crossovers where characters from different universes interact in unique ways. Luna's quirky and unique personality could lead to some interesting scenarios if Harry Potter were to act like her in an Avengers - related setting.
In such a fanfiction, Harry Potter, emulating Luna, would bring a touch of the magical and the eccentric to the Avengers' world. He'd probably view things from a different perspective, not as a typical hero but more like an otherworldly observer. He could befriend the Avengers in an unexpected way, perhaps by seeing through their tough exteriors to their insecurities. His actions might be based on Luna - like intuition rather than the more straightforward heroics of the Avengers. He could also introduce elements of magic that would completely baffle and intrigue the Avengers.
A third common theme is self - discovery. Harry, by acting like Luna, discovers new parts of himself. He might find that he has a more spiritual or intuitive side that he wasn't aware of before. This can be a journey of personal growth for him, and in the process, he might also help others around him to grow and change as well.
There's also 'When Harry Met Luna's World' which is quite interesting. In this fanfiction, Harry starts to act like Luna after an encounter with a strange magical artifact. He begins to see things from a different perspective, much like Luna does, and it creates some really funny and heart - warming situations in the story. For example, he starts talking to thestrals in the same way Luna might, which amuses his friends and confounds his enemies.
There could also be other characters from Hogwarts, like Neville or Ginny. Neville might be more accepting of Harry's change, as he has his own experiences with being different. Ginny could be in a unique position as she has a special relationship with Harry. She might be one of the first to notice the change in Harry and could have her own thoughts and feelings about it, either positive or negative.
One possible plot could be Harry starting to share Luna's unique views on magical creatures. For example, he might start believing in the existence of some of the more obscure creatures Luna is fond of, like the Crumple - Horned Snorkack. This could lead to him going on adventures to find evidence of these creatures, much like Luna would do.
You can start by looking on fanfiction websites like FanFiction.net or Archive of Our Own (AO3). On these platforms, you can use the search function and type in 'Harry Potter Luna' to find a wide variety of fanfictions centered around her character. Some fanfictions might explore her relationships, her adventures outside of the main story, or her unique perspective on the wizarding world.
A possible fanfiction could involve Lily Luna Potter and her time - traveling adventures. She accidentally stumbles upon a time - turning device and gets sent back to the time of the first Wizarding War. There, she has to be careful not to disrupt the timeline while also trying to learn from the events of that time to prevent a similar future catastrophe.
There could also be fanfictions that focus on Lily Luna Potter's family life. How does she interact with her parents, Harry and Ginny? What about her relationship with her siblings? Fanfiction might explore her growing up in a post - Voldemort wizarding world, dealing with the expectations that come with being the daughter of the famous Harry Potter.
Lily Luna Potter could play a key role in a fanfiction as a mediator. For example, in a situation where different wizarding factions are on the verge of war due to a misunderstanding, she uses her unique perspective, being part of the Potter family but also a new generation, to bring about peace and understanding.
There's a fanfiction that focuses on Lily Luna Potter's relationship with her family's legacy. She feels the pressure of being a Potter but also embraces it. In this story, she has to prove herself worthy of the Potter name in a magical competition held at Hogwarts. It's a story about self - discovery and living up to one's family's reputation.