There could be her personal diaries. Maybe she wrote about her time at Hogwarts, her friendships, and her early days with James Potter.
Perhaps there are some letters. Letters from her friends, from James, or from other important people in her life. These letters could reveal secrets about her life, her hopes, and her relationships that Harry never knew before.
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.
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.
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.
Sirius Black might be involved. As Harry's godfather and James' friend, he might have some knowledge about the trunk. He could help Harry understand the significance of the items in it. Also, Dumbledore could be involved. He always seemed to know a great deal about everything, and he might have some information about Lily's trunk that could help Harry on his journey of discovery.
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.
In the learning trunk, there might be a magical map similar to the Marauder's Map but with more detailed information about the Hogwarts library. It could show hidden sections and the location of books related to specific spells or magical theories. There could also be a set of magical crystals that when held, can transport him into virtual scenarios to practice spells and combat skills in a safe environment.
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.
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 could be Hagrid. Hagrid is always looking out for Harry and giving him things. He might come across this special trunk during one of his travels in the Forbidden Forest or in some wizarding market and think it would be perfect for Harry.
Hagrid could help. He has a lot of knowledge about magical creatures and objects. He might know where to find the right materials for the trunk, like some strong but flexible magical wood from a particular tree in the Forbidden Forest.