Maybe there is a misinformation or lack of communication in the fanfic. For example, the media or the people who record the heroic deeds in the story might misattribute his actions to someone else. So, Harry doesn't get the credit he deserves and doesn't realise that he is actually a hero. Also, if there are false rumors about him not being heroic, he might believe them and not see his own heroism.
It could be that he is constantly compared to other more 'flashy' characters in the fanfic world. Those characters might have more obvious powers or more dramatic backstories. So, Harry, with his more understated heroism, doesn't think he measures up and doesn't realise he is a hero too. He feels like he's just an ordinary person in the midst of these seemingly more extraordinary individuals.
In the fanfic, it might be that the people around Harry treat him as an ordinary person most of the time. So, he doesn't get the sense that he is a hero. For example, his friends might joke around with him and not give him the 'hero - treatment', which makes him oblivious to his heroic status.
A possible plot twist could be that the heirship is not what it seems. Maybe it's not related to a noble wizarding family but to an ancient, forgotten magic. Another twist could be that someone else has been manipulating events to make Harry think he's the heir when he's not, or vice versa.
It could be a vial made of enchanted glass. These are often used in the wizarding world to hold potions. It might be small and delicate, with a stopper that has its own magical properties.
He could have all sorts of wild adventures! Maybe he'd befriend some magical creatures and help them solve problems.
The author might use various methods. For example, they could keep key information from Harry's past hidden from him. Maybe they show his normal wizarding life at Hogwarts, with no signs of his heir status, so that when small hints start to appear, it creates mystery. His friends might not notice anything different either, which adds to the enigma.
He might meet a mysterious old wizard who has been traveling the seas for centuries, gathering knowledge of different magics. This wizard could be a great source of wisdom for Harry.
He could have a potions - making adventure. Snape might teach Harry some advanced potions techniques that are not taught at Hogwarts. Or Harry could discover some of Snape's old potions experiments.
Hermione would be a likely candidate. She's always been there for Harry and is very empathetic. She would probably offer him some words of wisdom and a shoulder to cry on.
Well, as Zelena's son, Harry could have her power over fire, which she sometimes uses in her magic. He could also have the ability to communicate with animals, as Zelena has a certain connection to the natural world. Another power he might possess is the power of teleportation. Zelena is quite mobile in the story, and Harry could inherit this trait, allowing him to move quickly from one place to another. But it's also possible that his powers are a combination of Zelena's and some unknown latent magic within him, which could make his powers unique and unpredictable.
The plot could develop with Harry secretly studying dark magic texts in the restricted section of the library. He might start to have strange dreams or experiences related to the dark magic he's exploring, and he has to be very careful not to get caught by Filch or the teachers.