Kind of generic, works better if you have more specific questions.
But, as it is a new event, here's some pointers.
The science behind this is aerodynamics, the basic principles behind Wright Stuff, Balloon Launch Glider, Helicopgter Duration & Gliders all broadly apply.
Keep it LIGHT. You are trying to minimize descent rate to reduce impact to the egg, that's the same as maximizing flight time. KEEP IT LIGHT. For everything that flies weight is critical to increasing flight time.
The rotor is a wing flying in a tight circle. This is the same thing as a propellor. See last years helicopter discussions for hints and tiops on rotor design and construction. You'll need to maximize rotor size to minimize descent, challenge will be to do so with minimum weight gain.
Like the other flying events you are trying to get the most flight out of limited energy. You need to really understand that your ONLY source of energy is the potential energy from your drop height. You have to use it efficiently if you want to maximize flight time.
Since you are starting from a complete stop, there is one big difference from helicopter rotor design. Lower pitch results in slower flights, but take longer to spin up. Higher pitch spins up faster, but doesn't slow as much. Depending on drop height, you may not reach full speed with a low pitch before hitting the ground so may still be dropping fast. Optimum pitch will be a tradeoff and require experimentation to figure out.
This a need to spin up also affects design. It takes energy (altitude) to spin up the egg, so why do it? You should have the rotor spin independant of the rest of the device.
So, start designing, start experimenting, and start modifying.
Oh, suggestion on the test process. You don't need to risk an egg every flight. Use an equivalent mass to the egg, similar shape for dynamics. Judge design by flight time and maybe speed (if you can measure it) at impact. Do most of your testing that way and then confirm with eggs when you are satisfied.