I should think Dill would also wish to save a memento of the night that Jem and Scout caused the mob of men to change their minds against going against Atticus and lynching Tom Robinson. Later Scout sees that Mr. Underwood has a double-barreled shotgun pointed at the men. Perhaps Dill would take a copy of the newspaper since, as Dill declares, "It's all over town this morning ...about how we held off a hundred folks with our bare hands."
Of course a memento of of the court house--maybe a fan-- where the trial for Tom is tried would be notable to Dill. And, something from the costume worn by Scout when she was attacked by Mr. Ewell.
This is a great and creative question. Dill Harris's summers were spent with the Finches, and when he runs away from home, he also winds up in Maycomb County with Atticus and family. Perhaps some items that could be included in a Dill Harris memory box could include his train ticket stubs to Maycomb, a lock of Scout's hair, some tiny trinket (porcelain figurine?) belonging to Aunt Rachel, sketches of Boo Radley, and maybe a few of the items that showed up in the knot hole of the Radley tree when Dill was actually present. Any of these items would serve to represent Dill well, and would serve as reminders of his time spent in Maycomb.
He may have movie tickets from Dracula. As Dill tells us, he lives with his mom in Meridian, Mississippi and is coming to spend every summer in Maycomb. Dill tells Jem and Scout of how his mother worked for a photographer in Meridian and had entered his picture in a beautiful child contest. His mom gave him the five dollars in winning and he had seen the Dracula picture show twenty times on the money. This statement almost immediately earns Dill the respect of Jem as he has never seen Dracula. How Dill brags about how he had won the contest and how he had seen the movie twenty times tells us how he may be a bit of a show off and likes to have something over everyone else.
Also possibly a book, as he is proud of his reading ability. The Grey Ghost is the book which he bet Jem if he touched the Radley house.