There's a fanfiction called 'Bond Beyond the Common' which is quite interesting. It focuses on how Luna's eccentricity, Hermione's intelligence and Harry's leadership skills combine. They go on a journey to discover a hidden part of the wizarding world and their bond is tested and strengthened in different ways throughout the story.
A possible fanfiction could involve them bonding over a shared love for a particular subject that's not part of the regular Hogwarts curriculum. Let's say it's the study of ancient and forgotten spells. Hermione is excited about the academic aspect, Harry is interested because he thinks it could help him in his fight against Voldemort, and Luna just has a natural curiosity. They start meeting secretly to research and in the process, their friendship blossoms.
One common theme is friendship. Writers often explore how the three characters' different personalities - Harry's bravery, Luna's eccentricity, and Hermione's intelligence - blend to form a unique and strong friendship.
In the fanfiction, Luna might be lost in a time - loop created by a malfunctioning time - turner. Hermione realizes what has happened due to her familiarity with time - turner magic. Harry goes with her to enter the time - loop. Inside, they face various challenges from different time periods. They find Luna, who has been trying to break the loop on her own. The three of them then work together. Hermione figures out the correct sequence of spells, Harry provides the necessary distraction to the time - loop's guardians, and Luna uses her natural connection to magic to help Hermione complete the spell that breaks the loop.
One idea could be a time - travel fanfiction where Harry, Hermione, Neville and Luna are sent back in time to the era of the Marauders. They could interact with James, Sirius, Remus and Peter, and perhaps change some of the events that led to the downfall of the Marauders. For example, they could prevent Peter from betraying his friends.
One way could be to focus on Hermione's insecurities. Maybe she notices Harry and Luna having deep conversations about the afterlife or strange magical creatures that she doesn't fully understand. She starts to feel left out and her jealousy builds as she sees them growing closer in a way that she's not a part of. For example, if Harry and Luna start going on little adventures to study some rare magical phenomenon without her, she might start to resent their new - found bond.
In these fanfictions, they often interact with great respect for each other. Hermione, with her logical mind, might initially be a bit skeptical of Luna's more out - there ideas, but she soon comes to appreciate them. Luna, on the other hand, admires Hermione's intelligence and determination. They might exchange ideas about magic, with Luna sharing her more esoteric knowledge and Hermione providing the more textbook - based facts.
One common theme is fixing past mistakes. Since they can travel through time, often they try to correct things that went wrong in the original timeline, like saving a character who died prematurely.
A really interesting one is 'Beyond the Golden Trio'. This fanfiction takes the relationship between Harry, Luna, and Hermione to a new level. It delves into their post - Hogwarts lives, showing how their friendship endures and grows, facing new challenges in the wizarding world outside of school.