Harry could decide to take the trunk to his room and study it more closely. He might call his friends over, like Hermione and Ron, to help him figure out what's inside and what the significance of the trunk is. They could use their combined knowledge of magic and history to try and unlock its secrets.
Perhaps he immediately senses some kind of dark magic associated with the trunk. So, instead of opening it right away, he goes to Dumbledore or some other trusted adult in the wizarding world to ask for advice on how to handle it safely.
He might first try to open it using some basic unlocking spells. If it doesn't work, he could search for a key around the attic.
Maybe it's filled with old family heirlooms like a dusty photo album with pictures of his ancestors he's never seen before, or some antique magic items that were long forgotten.
There could be some of his parents' old school things like their Hogwarts robes, wands (if they were allowed to keep them), and textbooks. Maybe there are also some love letters between his parents or photos of their time at Hogwarts.
It might make him see them more as real people. Before, they were just an idea of parents, but the trunk could contain things that show their personalities, like their sense of humor or their interests.
Finding the trunk may also make Harry more determined. If there are any signs of her fight against the dark forces in the trunk, like a plan she was making or a charm she was developing, it could fuel Harry's own determination to defeat Voldemort and protect the wizarding world.
Another possibility is that it just randomly appears one day. Magic in the Harry Potter world can be unpredictable. One morning, Harry wakes up and there it is at the foot of his bed. Or maybe a magical creature, like a house - elf that knew his mother, decides to bring the trunk to him. The house - elf could have been sworn to secrecy until the right moment when it senses that Harry is ready to discover his mother's secrets.
There could be her personal diaries. Maybe she wrote about her time at Hogwarts, her friendships, and her early days with James Potter.
Harry might have adventures like traveling to hidden magical realms with the genie or uncovering long-lost magical artifacts. It could be really exciting!
In the 'Harry finds Lily's school trunk' fanfiction, it could be that Harry stumbles upon the trunk and inside he finds a diary. Through this diary, he gets to read about Lily's adventures at Hogwarts, her friendships, and her early encounters with Snape. This would be a great way to explore Lily's character more fully from Harry's perspective.
Perhaps there could be her old Hogwarts textbooks. They might hold some special notes or spells that she had discovered during her time at school.
He might be shocked at first. Discovering that he isn't a Potter is like shattering his whole identity. He could feel lost and confused, not knowing who he really is anymore.