Obviously, Harry Potter himself is a main character. Since he is now the son of Hestia, his character development would be a central focus. He would be dealing with the implications of this new identity.
Yes, there are likely some fanfictions out there with this concept. You can start by checking popular fanfiction websites like Fanfiction.net or Archive of Our Own. Just search for 'Harry is the son of Hestia' in their search bars.
In this fanfiction, perhaps Harry would inherit some of Hestia's qualities. Hestia is the goddess of the hearth, so Harry might have a strong sense of home and family. He could be a more nurturing figure, different from his canon self. His magic might also be influenced by Hestia's domain, like having an affinity for protective spells related to the home.
In some fanfictions, there could be a plot where Harry, as the son of Hestia, has to face opposition from traditional wizarding families who don't accept this new 'parentage'. This would lead to him having to prove himself not only as a powerful wizard but also as a worthy heir to Hestia's legacy. He might have to use his wits and newfound powers to overcome these prejudices.
He would probably be more empathetic. Since Hestia is associated with the hearth and home, Harry would likely have a deep understanding of the importance of family and belonging. This could make him more sensitive to the feelings of others, especially those who have lost their homes or families.
The plot could involve Harry learning about his true mother. Hestia, being a rather different kind of mother figure compared to Lily Potter. His relationship with other characters might change as a result. For example, he might have a different connection to the Gryffindor common room which is associated with the idea of home, since Hestia has ties to that concept. He could also have some new abilities or a different magical aura that sets him apart from the original Harry Potter story.
One great fanfiction could be 'The Hestia - Bound Potter'. In this story, the fem Harry Potter as Hestia's daughter discovers her unique powers related to the hearth and home magic. It shows her journey through Hogwarts, facing challenges not only from the dark forces but also from the doubts within herself about her new identity.
One possibility could be that Hestia imparts her wisdom of the wizarding world to Harry in a very nurturing way. Since Hestia is the goddess of the hearth, she might create a warm and loving home environment for Harry, which is very different from his experience with the Dursleys. This could lead to Harry having a more stable emotional foundation as he grows up, and perhaps he would be less impulsive in his actions at Hogwarts.
Hestia Jones could be seen as a protector. If the lemon in the fanfiction represents something pure or valuable in the wizarding world, she would be the one guarding it. She would be brave and not afraid to stand up against those who would try to harm or steal the lemon - related thing. She might use her stealth and her knowledge of the wizarding world's hidden places to keep it safe.