You probably won't have time to completely investigate this, but you should definitely spend some time reading research papers. We can help you with finding appropriate references. Depending on the scope of your goals, you may want to work in a group. Look for projects where each person's work is separable, and yet everyone contributes toward a shared goal that could not be accomplished individually. Implement a class of more complicated primitives from Hanrahan's chapter in Glassner's book.
Fractals are relatively easy to implement and fun to use.
For extra fun, map textures onto your patches or fractals. Exotic wavelength-dependent effects such as dispersion and thin film effects. We can give you some references. Use any sample distribution, subdivision criteria, and reconstruction method you like. Allow interactive control over key parameters of your sampling scheme.