There are a number of different options for this IcP. Pick the one that fits in with what you are planning to do for your project. If none fits, do the
Search Tree (2-D Game) - If you have a game that has various things moving around in a 2-D world, you can use a tree to keep track of them. Technically this probably isn't needed unless you have several thousand or more, but you're going to write one in the name of learning even if you don't have that many things moving around.
Correlation Integral (Mathematica) - When you have a large set of points, an interesting function for them is the correlation integral. The normal definition of this function is scalar and depends only on distance. It can also be defined with distance along a direction in which case it can be scaled to higher dimensions. Calculating the full correlation integral requires O(N^2) time. However, if you limit yourself to a certain maximum range and use a spatial tree you can use the tree to only find pairs closer than the specified limit in roughly O(N log N) time. Add this functionality to your program. Make it so that you can load in or generate points of data. Then build a kD-tree with those points. Lastly, use the tree to calculate the values of the correlation function at different locations.
kD-Search Tree (Web Spider) - Another interesting use of a kD-tree is to help find related web pages. If you have a spider that can run through web pages, you have the ability to pull in all types of information. We want to find related pages based on word frequency. You will pick a set of words that you are interested in. Having 20-50 words is probably sufficient. The number of words is the dimensionality of your word frequency space. Every web page will be a point in this space. Words from the page that aren't on the list don't matter. For the words on the list, you count how many times each one occurs. The value you use for that dimension is math.log(1+count). Use that to build a kD-tree where every page has a point in your N-dimensional space. Look at the leaves of this tree. The pages sharing a single leaf node should have similar word counts. In order to make this effective you will need to spider a few thousand pages. Note that you don't have to store the whole page, just the link and the counts of the words you care about.
Using a Tree - If you can think of something else in your project that would benefit from a spatial tree, try it out.