There are many possible uses. As others have said, you can model a hypercube or something that makes use of a hypercube as well as modeling a change over time. However, there are many other possible uses.
For example, one of the theoretical simulation models of our universe uses 11th dimensional physics. You can write a program to model what these assumed physics would look like. The user would only be able to see a 3-dimensional space which definitely limits usability, but the 4th dimensional coordinate could act like the changing of a channel allowing the user to change their perspective. If a 4th dimensional explosion occurs, for example, you might even want a 5th dimensional array so that you can model what it looks like in each connected 3-dimensional space as well as how it looks in each frame of time.
To take a step away from the scientific, think about an MMORPG. Today many of those games uses "instanced" locations which means that a copy of a given zone is created exclusively for the use of a given group of players so to prevent lag. If this "instanced" concept was given a 4th dimensional coordinate and it allows players to shift their position across instances it could effectively allow all server worlds to be merged together while allowing the players a great deal of control over where they go while decreasing cost.
Of course, your question wants to know about ideas without using a GUI. That's a bit more difficult because you are working in a 2D environment. One real application would be Calculus. We have 3D graphing calculators, but for higher dimensions you pretty much have to do it by hand. A pogram that aims to solve these calculations for you might not be able to properly display the information, but you can certainly calculate it. Also, when hologaphic interfaces become a widespread reality it may be possible to represent a hypercube graph in 3D making such a program useful.
You might be able to write a text based board game where the position of pieces is represented with text. You can add dimensions and game rules to use them.
The simplest idea I could give you is a save state system. At each interval the program in memory is copied and stored into a file. It's coordinate is it's position in time. At face value you may not need a 4D array to handle this, but suppose the program you were saving states of used a 3D array. You could set it up to represent each saved state as a position in time that you can make use of and then view the change in time.
