Obviously, the main characters would be Harry and Tom's kids. But there could also be other characters like the teachers or fellow students who interact with them in the Great Hall.
It could center around the friendships or rivalries that their kids form in the Great Hall. For example, their kids could be sorted into different houses and have to deal with the implications of that. There could also be some sort of magical event that takes place in the Great Hall specifically related to their family's legacy. Their kids might have to face challenges based on the reputations of their fathers, Harry and Tom.
They might be curious about it. Since Tom Riddle is such a big part of Harry's past, his kids would probably want to know more. But it could also be a bit of a burden, like they feel they have to live up to something or prove they're not like Tom Riddle.
In a fanfiction, the three of them could be in the Great Hall for a sorting ceremony that goes wrong. A new student's magic is so powerful that it disrupts the normal process. Harry, Hermione, and Ron step in to calm things down. Hermione uses her vast knowledge of magic to try and control the situation, Ron's loyalty and bravery encourage the other students not to panic, and Harry's natural leadership helps them come up with a plan to contain the new student's magic until the teachers arrive.
You can start by looking on popular fanfiction websites like Archive of Our Own (AO3). It has a vast collection of fanfictions from various fandoms, and you're likely to find some great ones about Harry and Tom's kids there.
There's 'A New Generation's Alliance' which focuses on how Harry and Tom's kids come together to face a common threat. It delves deep into their individual personalities and how they grow to understand and trust each other over time. This fanfiction is great for those who like stories about friendship and cooperation between unlikely groups.
Their characters could also be developed through shared experiences in the Great Hall. For instance, they might face a new kind of dark magic threat in the Hall. Through this, Harry could show his leadership and Ginny her loyalty and fierceness, and they would learn more about each other and themselves as they overcome the challenge.
There was this fanfiction where a dark secret about Hogwarts was revealed. It was during a big feast in the Great Hall. Dumbledore stood up and announced that there was a hidden chamber in the castle that had been used for dark magic in the past. Harry, being the one who always felt responsible for the safety of Hogwarts, was devastated. He thought about all the times he had walked the halls, not knowing this danger. He started to cry silently. His green eyes were filled with fear and sadness as he looked around the Great Hall, at all the students who were now in potential danger because of this revelation.
He could be crying out of frustration. In the fanfiction, he might be facing difficulties in his studies or in learning new spells that are crucial for his fight against the dark forces. In the Great Hall, where he is constantly surrounded by students who seem to be doing better or at least not struggling as much as he is, he might feel inadequate and start crying. Also, if there are new threats emerging in the wizarding world and he feels like he's not making enough progress in preparing for them, this could lead to a breakdown in the Great Hall.
In a particular fanfiction, after a big battle, Harry and Hermione find themselves alone in the Great Hall. They start reminiscing about their years at Hogwarts, their first days, the friends they've lost, and how far they've come. It's a very emotional and heart - warming story that really delves into their relationship.