A theme that is often seen is the idea of adapting to new situations. The Scooby Gang has to quickly get used to the magic around them, and the Harry Potter characters have to deal with the Scooby Gang's unorthodox methods of investigation. This creates a lot of interesting scenarios where characters grow and learn from each other as they try to adapt to the new circumstances they find themselves in.
One great 'Harry Potter Fanfiction Scooby Gang Fanfic' could be 'The Mystery at Hogwarts'. In this fic, the Scooby Gang somehow finds themselves transported to Hogwarts. They team up with Harry and his friends to solve a mystery that involves both magic and some spooky elements that the Scooby Gang is used to. It's really interesting to see how the two groups interact and how the Scooby Gang's detective skills are put to use in the magical world.
One possible goal could be to protect the wizarding world from new and unknown threats. Since Harry has always been a defender, his gang would follow suit. Another goal might be to promote unity among different wizarding groups. In the original series, there were some divides, but as a leader, Harry could aim to bridge those gaps through his gang.
Harry's gang could include some of the characters we don't see much of in the original books. For example, the house - elves. Dobby was already very loyal to Harry. Maybe he could convince other house - elves to help out in a more active way. There could also be some of the older students who dropped out of Hogwarts for various reasons but still have a lot of magical knowledge. And of course, Hagrid could be an honorary member, providing his vast knowledge of magical creatures and his big - hearted support for Harry and the gang.
Harry himself would be one of the main members, of course. Hermione could also be part of it, using her vast knowledge of magic to the gang's advantage. Neville might join too, having proven his courage time and time again. And perhaps some of the Weasley twins' friends from their pranking escapades.
The plot might involve Harry being part of a gang that is formed to protect the Muggle - born students. They operate in the shadows, using their combined skills to counter any threats from the pure - blood supremacist groups. They could be led by an older student who has seen the discrimination first - hand and is determined to stop it.
Dumbledore would be a very wise and calming presence in any interaction with the Scooby Doo gang. He could offer guidance and advice as they try to solve mysteries in the wizarding world. Meanwhile, Scooby might accidentally lead the Hogwarts students to some hidden passages or secrets in the castle because of his nose for trouble. And the ghosts in Hogwarts could either be really helpful to the Scooby Doo gang, like Nearly Headless Nick guiding them around, or be mistaken for the villains they usually chase, causing some comical misunderstandings.
You'll often find Draco Malfoy in these fanfictions too. He could be the leader of a rival gang, with his Slytherin cronies. Neville Longbottom might also be a character, perhaps having transformed into a tough and resourceful member of a gang, using his knowledge of Herbology to create magical defenses for the gang. And of course, Dumbledore could be a sort of wise figure in the background, guiding Harry or other protagonists from afar.