One project idea is to design a navigable map of Gotham City. The city, which is said to be inspired by American metropolises such as New York City and Chicago, is a huge place, but if you read through the comics and watch the Batman movies, you'll get a good feeling about where things are located. Make a map that labels the boroughs of the city along with important locations like Arkham Asylum, Wayne Manor and other places that show up in the comics.
While DC comics has updated Gotham City as the world we live in has grown more technologically advanced, you can find interesting tidbits about the history of the city by reading the source material . Use a timeline to document roughly when Batman first debuted, when his villains began appearing, and when important events transpired in the city.
Unlike Marvel Comics, which uses many real cities and places in its fictional stories, DC Comics uses fictional cities like Metropolis and Gotham City. An interesting project might be to compare Gotham City to real American cities and see which ones it likely drew inspiration from and which city Gotham is best compared to in the modern day. Looking at weather patterns, crime demographics, nightlife and other factors such as architecture are all good comparison points for making a case as to where Gotham City "should be" in reality.
Gotham City plays host to a bevy of famous people, from costumed vigilantes to insane super-criminals. Drawing up a walk of fame for Gotham City would include famous fictional figures such as Batman, Robin, the Joker and the rest of the rogues gallery, of course, but it should also include other important figures. Cops like James Gordon and Harvey Bullock, for instance, along with average criminals like Carmine "The Roman" Falcone and the other organized crime families that rub shoulders with Batman in important plot lines in the comic books.