In the fanfiction, perhaps Stain trains Izuku because he believes Izuku has the purity of heart to use power in a way that Stain deems correct. Stain could teach Izuku how to analyze opponents better, which would enhance Izuku's ability to strategize in battles. Izuku might then start to question the current hero ranking system during his training. He could grow to be a hero who is more focused on the essence of heroism rather than just fame and recognition, which would make him stand out among his peers and also cause some friction with the more traditional heroes and hero agencies.
One popular theme is exploring Stain's ideology further. Many fanfictions dig deeper into his beliefs about true heroes and villains, and how that might play out in different scenarios.
Another place to check could be FanFiction.net. It has a large collection of fan - made stories for various series, including My Hero Academia. However, the quality can vary, so you might need to sift through a bit to find the really good Stain fanfictions.