Another possibility is Luna Lovegood. In some fanfictions, Luna has hidden powers or knowledge. She could be the one to introduce Harry to legilimency in a more unconventional way, perhaps through her unique perspective on magic and the wizarding world.
Dumbledore could also be the mentor. He has great knowledge and wisdom, and he might see the potential in Harry learning legilimency. He could guide Harry in using this power for good and help him control it so that he doesn't misuse it.
Snape is a likely candidate. He is a skilled legilimens in the original series, and his relationship with Harry is already complex. Teaching Harry legilimency could be a way for Snape to either help or hinder Harry, depending on his motives.
It could be an older relative who was thought to be dead but actually has been living in seclusion, waiting for the right time to teach Harry about their family magic. This relative might have a vast knowledge of the family's magical traditions and spells.
It could be a powerful wizard from Asia who has long studied both magic and kung fu. This wizard has decided to pass on his knowledge to Harry Potter, seeing his potential.
One interesting plot point could be Harry's initial struggle with legilimency. Maybe he has a hard time controlling it and accidentally invades his friends' minds, causing some awkward or tense situations.
Another reason is that it allows for a lot of creativity. Writers can create new mentors for Harry to learn from, new techniques of Legilimency, and new consequences of using it. It's a topic that can be combined with other elements of the Harry Potter world, like the prophecy, the Horcruxes, and the wizarding politics, creating a rich and engaging fanfiction.
It could be Hagrid. He has a lot of knowledge about different magical creatures and skills. He might introduce Harry to marksmanship as a way to protect himself better in the wild. Hagrid could show Harry how to make his own arrows and teach him the basics of aiming and shooting.
One popular 'Harry Potter Learns Legilimency' fanfiction is 'The Mind's Eye' on Fanfiction.net. It has an interesting take on how Harry discovers and masters Legilimency with the help of an unexpected mentor.
It could be Dumbledore, even if he's passed away. His portrait or some kind of magical remnant of him might still be able to guide Harry in learning new magic.
Perhaps Hagrid could be his mentor. Hagrid has a lot of knowledge about magical creatures and their parts, which could be useful for Harry in his blacksmithing. He could show Harry how to use things like dragon scales or unicorn hairs in his creations.
In Harry Potter fanfiction, Legilimency often takes on new and exciting interpretations. Some fanfics might explore how characters develop their Legilimency skills further. For example, a character could use Legilimency to uncover hidden secrets about the wizarding world's history that were not revealed in the original books. It could also be used in unique ways during battles, like reading an opponent's next move in a duel.
Dumbledore could be his mentor. He has vast knowledge and experience, and he would likely see the potential in Harry being an Animagus. He could guide Harry through the difficult process of learning to control his transformation and use his Animagus form for good.