It could be stronger in certain dark arts. Given his parentage, he might have a natural affinity for curses and hexes that are associated with the dark side. He could learn and perform these spells with greater ease compared to other wizards. However, he might struggle with defensive and positive magic, like healing spells or protective charms.
He would probably be more conflicted. On one hand, he would have the potential for great good, like the original Harry. But on the other hand, he would have the shadow of Bellatrix's dark deeds hanging over him. This internal conflict could make him more withdrawn at times. He might also have a fascination with the dark arts, not in an evil way, but more out of a need to understand his mother's world. This would set him apart from the Harry we know, who was always more firmly on the side of light magic.
Well, Voldemort's magic was mainly focused on dark and destructive spells. His son, on the other hand, could have a more balanced magic. Maybe he has an affinity for defensive spells as a way to distance himself from his father's aggressive magic. He might also have a unique ability to heal, which would be a direct contrast to Voldemort's harm - causing magic.
His magic could be more intense and have a different flavor. The influence of Aphrodite might make his spells more alluring. For example, a stunning spell could have a sort of charming effect added to it, making the victim more docile. From Ares, his magic could be more aggressive, like his blasting spells could be more powerful and destructive.
Harry could become more secretive and distrustful. Growing up with Bellatrix, who is part of the Death Eaters and has a lot of secrets, Harry would learn to keep things to himself. He may not trust others easily, even those who seem friendly, as he has been brought up in an environment full of lies and hidden agendas.
One common theme might be identity crisis. Harry would be conflicted about his dark heritage and his place in the wizarding world. Another could be power struggle, as he may inherit some of Voldemort's power.
Bellatrix's character would have to be rewritten to some extent. Saving Harry would go against everything she has stood for in the original story. So, she might become more conflicted, showing both her old, evil self and a new, emerging self that is capable of good. Her relationships with other Death Eaters would also be affected, as they would view her actions as betrayal. This would add a whole new layer of drama and complexity to her character in the fanfic.
His magic could be different in terms of its source. In Harry Potter, magic seems to be more inborn and related to one's heritage and the magic within themselves. In Fairy Tail, magic can be learned and developed in different ways, like through training and guild support. So Harry, raised by Fairy Tail, might have a broader understanding of where his magic comes from. He might see it as something that can be continuously improved and not just as a natural gift. This could lead to him constantly seeking new ways to enhance his magic, perhaps by using Fairy Tail's training methods on traditional Hogwarts spells, which could result in some really unique magical effects.
One plot idea could be Harry growing up in a very different environment. Instead of the Dursleys, he's raised in the dark wizarding world under the influence of Bellatrix and Voldemort. He might struggle with his identity, having the blood of both a hero (in a way, as he's related to Harry Potter in the original story) and two of the darkest wizards. He could be trained in dark magic from a young age but have an inner conflict due to some latent goodness in him.
Harry might be more secretive. He'd have to hide any doubts or conflicting feelings he has about the dark path he's being led down. Bellatrix would not tolerate any signs of weakness or deviation from her plans. So, he'd become very good at hiding his true self, which would make his personality complex and full of contradictions.
His character development would be extremely different. He would no longer be the 'chosen one' in the traditional sense. He might feel a sense of shame or guilt that he didn't have before. His self - confidence would take a huge hit. He would constantly be looking over his shoulder, wondering if people were judging him based on his father. His magic might also change. He could have an easier time with some of the more complex and darker spells, but at the same time, he would have to be extra careful not to let that power corrupt him. His moral compass would be all over the place as he tries to reconcile the fact that he is the son of the most evil wizard with his own desire to be a good person.