In some fanfiction, a possible plot could involve Harry using his time in the orphanage to study magic on his own. He might find some old, tattered magic books in the orphanage attic that were left behind by a former wizard who lived there secretly. The orphanage could also be a place where dark wizards are secretly observing Harry, waiting for the right moment to use him for their own evil purposes. Harry would then have to use his wits and the little magic he knows to outsmart them. Another aspect could be that the orphans' collective emotions create a unique magical force that Harry has to learn to control and use for good.
Well, a cool plot could be that Harry forms a strong bond with an old caretaker at the orphanage who turns out to have a mysterious past in the wizarding world. This caretaker could be the one to introduce Harry to some basic magic concepts before he even gets his Hogwarts letter. Also, there could be a rival orphan who is jealous of Harry's emerging magic and tries to sabotage him. And what if the orphanage is a gateway to a parallel wizarding world that has been cut off from the main one?
One plot idea could be that Harry discovers a hidden magic within the orphanage itself. Maybe there's a secret room that holds ancient spells or a connection to a long - forgotten wizarding family. Another idea is that the other orphans start to show signs of magic too, and Harry has to figure out how to protect them from the wrong kind of attention. Or perhaps the orphanage is under a curse, and Harry's presence there is the key to breaking it.