A unique way for their relationship to grow is if they are both affected by a strange scientific experiment at Oscorp. This gives them a shared experience that no one else can understand. They have to rely on each other to figure out what's happening to them and how to reverse the effects if they're negative. This would bring them closer together on a very personal level.
Their relationship could develop uniquely if Harry becomes a sort of mentor to Peter in the business and science side of things. Since Harry has access to Oscorp's resources, he could teach Peter about advanced technology that could be useful for his Spider - Man activities. This would create a different dynamic from the typical superhero - sidekick relationship.
Maybe Harry, who is initially jealous of Peter's popularity as Spider - Man, realizes that he can have a different kind of impact on the city. He starts to work behind the scenes, using his wealth and influence to support Peter's crime - fighting efforts. This way, their relationship evolves from one of potential envy to one of mutual respect and cooperation.