Well, one possibility is that Draco would be completely out of his element at the Dursleys'. He's used to luxury and magic, while the Dursleys' house is a non - magical and rather dull place. He might start to understand Harry's pre - Hogwarts life better. For example, he could see how Harry was treated like an outcast, which might change his view of Harry. Maybe they'd bond over their shared dislike of the Dursleys' rules and behavior towards them.
Well, one possibility is that Draco starts to see the value in the Dursleys' family structure, despite its flaws. He could form an unlikely bond with Dudley, and they might teach each other things. Draco could show Dudley some magic tricks in secret, while Dudley could introduce Draco to Muggle games like football. It could be a really interesting exploration of two different worlds colliding within the Dursleys' household.
Draco would surely interact with the Dursleys. Vernon would probably be very suspicious and hostile towards Draco at first, seeing him as another strange and magical being. Petunia might be a bit more reserved but also curious in her own way. Dudley could range from being intimidated by Draco's wizarding background to being fascinated. And don't forget about Hedwig! If she's still around, Draco might have some sort of interaction with the owl, which could be really funny or interesting as he tries to figure out how to deal with this magical creature in the Muggle - dominated Dursleys' household.
Maybe it was an accident. Like a wrong Floo Powder destination or some kind of misfired spell that landed Draco right at the Dursleys' doorstep just as Harry was there.
She might drive the Dursleys crazy with her intelligence and magic knowledge. For example, she could accidentally use magic in front of them while trying to solve some household problems, which would surely shock them as they are so against magic.
If Snape stays at the Dursleys in fanfiction, it could lead to some interesting character interactions. Snape might have a complex relationship with Dudley. He could try to teach Dudley a lesson about bullying, using his Slytherin - like cunning. Or perhaps he'd end up having some sort of begrudging respect for Vernon Dursley's determination to keep the wizarding world out of his family's life, while at the same time finding it completely absurd. And with Petunia, there could be a whole range of emotions, from her fear of him as a wizard to a possible shared understanding of their complex feelings towards Lily Potter.
Well, it could be that Draco has a change of heart early on. He realizes that the Dursleys are truly awful to Harry. So he uses magic to create distractions. He could make strange things happen around the house that scare the Dursleys so much that they just want Harry to leave. Then Draco could be waiting outside and whisk Harry away to a safer place, like a hidden place in the wizarding world where he has set up a little haven for Harry.
Maybe Draco finally understands that Harry is a victim of circumstances. The Dursleys' treatment of Harry is cruel, and Draco, having grown up in the wizarding world, realizes it's wrong. So he steps in to rescue Harry.
He could be doing it out of a sense of justice. Despite his family's views, Draco might see that the way the Dursleys treat Harry is wrong and decides to step in.
It could be part of a plot where Draco is given an assignment by someone in the wizarding world to save Harry from the Dursleys. He initially does it for self - serving reasons like getting in good with certain powerful wizards who are concerned about Harry's well - being. But during the process, he actually starts to care about Harry.