One example could be a fanfiction where Harry discovers his cat familiar has the ability to sense dark magic. The cat might help him in his adventures by warning him of approaching danger.
There might be a fanfiction where the cat familiar is a magical cross - breed. It could be part Kneazle, which gives it special intelligence and loyalty, and part something else that gives it unique powers like the ability to turn invisible for short periods. This cat could play an important role in Harry's life, for instance, helping him to find hidden passages in Hogwarts by using its unique senses.
One example could be a dark familiar in the form of a shadowy wolf that follows Harry Potter. It might have powers related to stealth and be able to sense danger in the wizarding world. Another could be a raven with a mysterious aura, perhaps having the ability to communicate telepathically with Harry.
I haven't actually read this specific fanfic, so I can't give exact plot examples. But it could be something like Harry is turned into a cat (a common pet familiar) and has to navigate Hogwarts in that form, trying to communicate with his friends without being discovered.
One example could be a fanfiction where Harry Potter has an augment that allows him to communicate with magical creatures in a more profound way. This could lead to him forming alliances with creatures like centaurs and house - elves in a different manner than in the original series, and they could help him in his fight against Voldemort.
One plot could be that the robot is a gift from Dumbledore to Harry. It's designed to protect him from Voldemort in a way that magic can't. So, it has special abilities like detecting dark magic from afar and alerting Harry. Another plot might be that the robot is an experiment gone wrong from a wizard who was trying to combine magic and Muggle technology. It then finds its way to Harry, and he has to figure out how to deal with it and keep it a secret from the Ministry.
One possible plotline could be that Harry Potter decides to take on the Dark Arts in a more proactive way. Instead of just reacting to Voldemort's attacks, he studies the Dark Arts himself to find weaknesses, which is a very bold move for him considering the taboo nature of the Dark Arts in the wizarding world.
In some fanfictions, the color changing contacts might be a gift from a magical creature. For instance, a friendly house - elf gives Harry the contacts. These contacts could change color according to Harry's mood. So when he's angry, they turn red, and when he's happy, they turn a bright blue.
One popular 'Harry Potter has a familiar' fanfiction is 'The Familiar's Bond'. It delves into Harry having an unexpected familiar that aids him in his adventures at Hogwarts. It adds an interesting twist to the original story.
Some fanfictions that are overly long - winded and full of filler content can be considered the worst. They might have pages and pages of a character just thinking about nothing important or going through mundane daily activities that have no real connection to the main plot of the Harry Potter world.
One example could be a fanfic where Harry has the ability to control time without any limits, which is much more powerful than the time - turning devices in the original series. Another might be a story where Voldemort is made invincible and can defeat any opponent with a single spell, completely overpowering the other characters.
One example could be 'The Dominant Potter' where Harry discovers a new and extremely powerful form of magic early on and uses it to take charge in the fight against Voldemort. He is shown as being dominant in strategic planning and combat situations.