Perhaps it's a side character from the original books who decides to strike out on their own for some reason. For instance, Neville Longbottom could be reimagined as the lone traveler. After facing so much in the main story, he might embark on a personal journey to discover more about his family's history and magic, all by himself.
It could be an original character created by the fanfic writer. This new character might be a transfer student to Hogwarts who doesn't quite fit in at first and thus often travels alone. They might have a mysterious past that makes them keep to themselves.