They might choose to run away because they want privacy. After all the battles and saving the world multiple times, they just want some time alone without any interruptions from other demigods or the gods. So running away seems like a good option.
Perhaps they run away because they are tired of all the rules and expectations placed on them. They just want to be free and explore the world on their own terms. They could end up in a small, idyllic town where they learn more about normal human life and each other away from the chaos of the demigod realm.
He could run away because he feels pressured. When he meets Annabeth, it's important because she can offer him support and guidance. In fanfiction, their meeting can lead to many different plot developments. For instance, they might go on a quest to find a lost artifact that has the power to change their world. Annabeth's knowledge and Percy's courage will be crucial in this journey.
One reason could be to avoid a dangerous prophecy that they believe the camp can't protect them from. In some fanfictions, they might think that by running away, they can face the threat on their own terms.
In this fanfiction, Percy and Thalia could meet a powerful sorcerer who has been exiled from the magical community. The sorcerer could see potential in them and offer to teach them some ancient magic spells that could be useful against their enemies. Additionally, they may come across a group of forest spirits. These spirits could be initially hostile but after learning of Percy and Thalia's noble intentions, they could provide them with shelter and information about hidden paths that lead to safe places away from the threats they are running from.
They might choose to run away secretly because they want to be alone together. They are in love, and the hustle and bustle of Hogwarts life doesn't give them enough privacy. So, they run away to a quiet place where they can enjoy each other's company without any interruptions.
Perhaps they are tired of the constant power struggles and drama. Bella has seen a lot of chaos in the vampire world, and Jasper has been through it for a long time. Together, they decide to run away to find some peace. They want a life away from the battles, the territorial disputes, and the never - ending family feuds. They believe that by running away, they can start anew and build a more harmonious existence for themselves.
They could meet other demigods who are also on the run or lost. These demigods might have different powers and backgrounds, and together they could form a sort of 'runaway group'. For example, they might meet a demigod who has the power of controlling plants and can help them find food and shelter in the wild.
She might take it because she loves the smell of it. It has Percy's scent, which makes her feel close to him.
It could be Chiron, the centaur. He's always keeping an eye on the campers and would likely be concerned if he caught them up to something. He might give them a stern talking - to and some extra chores as punishment.
Some fans like it because it offers an alternative to the canon relationships. It gives a chance to explore a different dynamic between Harry and Hermione, one that might be more romantic or at least a very deep friendship that's different from what's shown in the books. It allows for creativity and new storylines.