One possible plotline could be Harry growing up in an American wizarding community that has different traditions and values compared to the British one. He might have to adapt to new spells and magical practices.
In certain fanfictions of this type, Harry could be found by a powerful but reclusive wizard. This wizard decides to mentor Harry in the ways of ancient and lost magics. Harry then uses this knowledge to not only become a very powerful wizard but also to change the wizarding world's view on things like blood purity. He may also go on a journey to find out why the Dursleys abandoned him and in the process, uncover some dark secrets about his family that had been long buried.
One possible plotline could be Harry finding a new family among the magical creatures. For example, he might be taken in by a friendly centaur herd who teach him their ways of magic and survival. Another could be Harry being found by a secret group of wizards who are against the Dursleys' cruel treatment and they help him develop his magic in a more positive and nurturing environment. And there could also be a plot where Harry discovers a hidden magical power within himself as a result of the abandonment, like the ability to communicate with the spirits of Hogwarts, which then guides him on a new path to face Voldemort.
A third plotline might be that Harry is abandoned in a magical but hidden place. There, he is raised by magical creatures, say, a group of centaurs. They teach him their own brand of magic and wisdom. When he finally enters the wizarding world proper, he has a very different perspective on magic compared to the other students, which causes both admiration and some envy among his peers.
One possible plotline could be Harry growing up in an orphanage where he discovers some latent magic powers on his own before getting his Hogwarts letter. He might face more difficulties and prejudices from the muggles around him, which could make his entry into the wizarding world even more of a relief.
One interesting aspect could be the unique plot ideas that were left unfinished. For example, some might have had really creative takes on the wizarding world's history or new magical creatures. Another aspect is the different character developments that were being explored. Maybe a fanfiction was showing a side of a character we don't see much in the original series. Also, the writing styles in these abandoned works can be fascinating as they often vary from very formal to extremely creative and unorthodox.
Well, some might involve magical creatures with furry characteristics in new adventures at Hogwarts. For example, a werewolf character could have a unique story arc that intersects with the main Harry Potter plot. Maybe they are trying to find a cure or dealing with the prejudice within the wizarding world in a different way than what's in the original books.
One possible plotline could be a comical take on baby Harry's magical mishaps. Maybe he accidentally makes his diapers levitate or turns his crib into a mini Hogwarts. Another could be about how the other characters like Hagrid or Dumbledore interact with baby Harry in this unique situation, perhaps trying to figure out how to handle his early magical outbursts while he's still in diapers.
Perhaps a plotline where the ogre has some sort of special power that the wizards in Harry Potter are interested in. This could lead to a whole series of events where they try to either control or cooperate with the ogre.
In some fanfiction, a familiar might get lost in a different dimension. So, the main character has to go on a journey to find it. Along the way, they meet strange new creatures and learn new spells, which is really exciting. It also shows a different side of the bond between a wizard and their familiar.
There could also be a fanfiction where a member of the Sacred 28 discovers a long - lost family secret that has the potential to change the power dynamics among the pure - blood families. This secret could be related to a hidden magic or a forgotten alliance, and as they try to uncover more about it, they face opposition from other families who fear the change it might bring.