Sirius Black's family (before the whole dark turn). Imagine if Harry was fostered by the Blacks in a world where they hadn't gone down the dark path. He could have learned a lot about pure - blood wizarding traditions in a more positive light, and also had a family with a strong magical lineage.
One idea could be that Harry is placed with a family of magical creatures protectors. They teach him about the hidden magic in nature and the importance of protecting all magical beings. This could lead to Harry having a different view on magic compared to what he learned at Hogwarts.
In a fanfiction, Foster could be a descendant of an ancient wizarding family with a connection to the Bifrost. Harry Potter, while on one of his adventures, stumbles upon Foster. Together, they embark on a journey to the Bifrost. Along the journey, they learn about the history of their respective families and how they are intertwined with the magic of the Bifrost. They also face challenges like dark wizards who want to control the power of the Bifrost for evil purposes.
Foster could be a Muggle - born who has an innate ability to sense the Bifrost. Harry Potter, being the famous wizard he is, helps Foster understand and control this new - found connection. They might use the Bifrost to travel to different magical realms and discover new spells or magical artifacts. For example, they could end up in a realm where magic is based on music and have to learn to adapt to use their magic there.
One possible fanfiction could involve Jane Foster using the Bifrost to accidentally end up in the Harry Potter universe. Maybe she arrives at Hogwarts and is mistaken for a new kind of magic user. The students and professors would be intrigued by her Asgardian technology and different kind of magic. She could potentially teach them about the Nine Realms while also learning about the wizarding world's magic.
The foster family characters can be developed in many ways. If they are good - hearted, they could show Harry different values. They might be very strict about magic use, which makes Harry respect the rules more. Or if they are bad, they could have their own motives for taking Harry in, like using him for his fame. Their actions and words can gradually reveal their true natures.
Harry Potter and Jane Foster themselves are great to pair. Their different backgrounds, one in magic and the other in science, can create a lot of interesting interactions. Hermione could also be paired with Jane Foster. Hermione's thirst for knowledge and love of learning could mesh well with Jane's scientific mind. They could exchange ideas about different forms of knowledge. Another good pair could be Thor (since he's related to Jane Foster) and Hagrid. They both have a big - hearted and strong - willed nature and could have some fun adventures together in a fanfic.
Well, a fanfiction could involve Jane Foster being a healer at Hogwarts. Since she has some medical knowledge from her own world, she could introduce new ways of treating magical injuries. Or, Harry Potter could be sent on a mission to Jane Foster's world and he has to adapt to the different technology and culture there while still using his magic to solve problems. Also, there could be a story where the Asgardian magic in Jane Foster's world has some sort of connection to the magic in the 'Harry Potter' universe, and they have to figure out how to use this connection to save both worlds from a great evil.
A common theme is the exploration of different power sources. Harry has magic and Jane has her scientific knowledge. Another could be culture shock, as they come from very different fictional cultures.
Yes, there are some out there. You can start by looking on popular fanfiction websites like Archive of Our Own. Just search for 'Harry Potter x Jane Foster' and you'll likely find a variety of stories with different plotlines and interpretations.