One interesting story could be where Hermione takes it upon herself to study Buckbeak's unique behaviors more closely in Hogwarts. She might discover new magical aspects related to his species.
There's 'The Adventures of Harry, Hermione and Buckbeak'. It has a really interesting plot where they use Buckbeak to explore ancient wizarding ruins. There are lots of cool details about the ruins and how they interact with the environment around them using magic.
Another place could be FanFiction.net. There are many Harry Potter - related fanfictions on this site, and you're likely to find some great ones about Hermione and Buckbeak. Just make sure to use the search function effectively.
One interesting 'Harry and Hermione Buckbeak' fanfiction could be where they take Buckbeak on a secret adventure through the Forbidden Forest. They might discover new magical creatures and have to rely on Buckbeak's strength and their own wits to get out of tricky situations.
At first, it might be a bit tentative. Harry has to earn Buckbeak's trust, just like in the original story. But as time goes on, they start to understand each other better. Buckbeak realizes Harry is different from other humans who might harm him, and Harry sees Buckbeak as more than just a magical creature.
One common theme is freedom. Both Buckbeak and Sirius have been wronged and are in a fight for their freedom. In many fanfictions, they help each other achieve that freedom, whether it's Sirius using Buckbeak to escape or Buckbeak being freed with Sirius' help.
One great Buckbeak and Sirius fanfiction is 'The Flight of Freedom'. It imagines Sirius bonding with Buckbeak during his time on the run. It beautifully depicts how Buckbeak could sense Sirius' innocence and loyalty, and they form an unbreakable bond as they evade the Ministry's capture. It's full of adventure and heartwarming moments.
The main characters are clearly Harry Potter and Buckbeak. Harry is the famous wizard boy from Hogwarts, and Buckbeak is the hippogriff that Harry befriends in the series.
There might be a fanfiction where Buckbeak plays a crucial role in protecting Hogwarts. He could use his sharp eyes and powerful wings to spot incoming threats before the students and teachers are aware. For example, he might notice a group of Death Eaters approaching and warn the school in time. He could also carry important messages between different parts of the school or even outside of Hogwarts to get help when needed.