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.
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.
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 possible plot would be that Jane is researching something related to energy or space - time when she stumbles upon a magical rift. Out comes Harry. At first, they are both confused. But as they explore their family history, they find out they are siblings. Then they have to deal with the fact that there are forces in both their worlds that don't want them to be together. They would go on a journey to find out why and how they are related and also to protect their respective worlds from those who oppose their union.
Jane Foster could introduce Harry Potter to the concept of the Bifrost. She might show him how it works and how it can be used to travel between different realms. Harry, in turn, could share his knowledge of magic with her. They could then use their combined knowledge to go on an adventure, perhaps to a new and unknown realm.
For this fanfiction idea, perhaps there was a hidden connection all along. Maybe a powerful wizard or an ancient Asgardian magic was involved in their family's past. Harry's life at Hogwarts would be very different with a sister like Jane Foster. He could learn about different dimensions and technologies from her. And Jane, on the other hand, could be exposed to the wizarding world's secrets. They could have adventures together, using both magic and science to overcome challenges.
In fanfiction about Doctor Strange and Jane Foster, their relationship could be a complex one. Since they come from different worlds - Strange from the world of magic and Jane from the world of science. However, they may be drawn to each other because of their similarities too. They are both highly intelligent and dedicated to their respective fields. Fanfic writers might explore how they deal with the cultural differences between the magical and scientific communities while trying to understand each other on a deeper level.
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.