He could gain the ability to combine magic with physical strength. For example, he might be able to use spells that enhance his already powerful punches or kicks, like Thor's strength - based attacks. Also, he could have a better understanding of elemental magic, perhaps being able to control lightning like Thor does, but in a magical way.
Well, Harry Potter might gain some extraordinary magical abilities from being raised by Thor. His magic could become more instinctive and less reliant on incantations. Just as Thor wields his power without much thought in the heat of battle, Harry could do the same with his magic. He might also gain the ability to sense magic in a different way, perhaps more like how Thor senses power. His ability to teleport or apparate could be enhanced and combined with Thor's ability to travel through different realms. This would make Harry a much more powerful wizard, able to traverse not only the wizarding world but also other magical or non - magical realms with ease.
If raised by Newt, Harry's ability to sense the emotions and needs of magical creatures would be enhanced. This is because Newt has a great understanding of these aspects. Harry might also develop unique magical abilities that are specific to interacting with the habitats of magical beasts, like being able to create or modify suitable environments for them with magic.
He might gain enhanced spiritual powers. Kami could teach Naruto to better sense and interact with spiritual energy, which could give him an edge in battles against opponents with spiritual - based attacks.
He could gain better stealth skills. Hidan is quite good at sneaking around and Naruto might pick up on that. So he could move more quietly and be better at surprise attacks.
Harry might gain the ability to dodge attacks more easily. Sans has a certain 'instinct' for avoiding harm, and if Harry had this, it would be much harder for his enemies to land a hit on him. This would be really useful in duels, especially against powerful dark wizards. It could also make his Quidditch playing even more amazing as he could dodge bludgers effortlessly.
His magical abilities would likely develop at a much faster rate. The founders are extremely powerful and knowledgeable. They could guide Harry to discover and master magic that most wizards his age wouldn't even know exists.
He gains super - speed, of course. This allows him to move faster than any normal wizard. He can also vibrate his molecules to pass through solid objects, like walls in Hogwarts, which gives him a great advantage for sneaking around or escaping from tight spots.
In fanfiction, Harry Potter's magical abilities would offer a lot to the Avengers. For one, his magic can create various objects out of thin air, which could be useful in creating distractions or providing necessary tools in the middle of a battle. His magical knowledge of potions could also potentially be used to create new substances that could enhance the Avengers' powers or heal them more quickly. And his ability to communicate with magical creatures might come in handy if there are any strange or otherworldly beings that the Avengers need to interact with.
His abilities in alchemy would be far more advanced. He'd likely be able to create potions and elixirs with ease, having learned from Flamel. Also, his understanding of magical objects, especially those related to the Philosopher's Stone, would be greater.
His magical abilities might develop with a greater emphasis on intellectual magic. Since Rowena Ravenclaw is associated with knowledge and wisdom, Harry could be more inclined towards spells that require deep understanding and study. For example, he might master complex transfiguration spells earlier because Ravenclaw would encourage him to study the theory behind them thoroughly.
His magical abilities might be influenced by the Norse magic system. He could have a more primal and elemental approach to magic, perhaps being able to control natural forces like thunder (associated with Thor) or ice (linked to some Norse entities).