how sophisticated do you want to be? well, if it is for an introductory physics class, maybe I can help you.
suggestions:
1. moment of inertia of raw egg and boiled egg.. you can conduct an experiment yourself.
2. the structure and the shape of the egg. How much stress can the egg sustain when some forces are applied on different parts of the egg.
3. the center of mass of the egg... kind of stupid, but just an idea.
4. density of the egg. raw egg vs. boiled egg. Does it float in water, sink in water...etc. conduct an experiment measuring the density of eggs using water and salt (if the egg sinks). if the egg floats... then, use oil and soap (that would be messy, hehehe) or just mass the egg and measure the volume (the most usual way).
5. how much heat energy it takes to cook an egg.
6. can egg be frozen?
that's all I can think of right now...