The Quadrotors I've worked with have all been built as light as possible, have an arduino/picoitx control system, and fly for around 10-15 minutes.
Building one out of FRC-sanctioned parts is an interesting challenge, but I have to ask, why? If you ever manage to get it off the ground, I'd be worried about safety, first off. Sure, you can fly it on a tether, but when a large, heavy object is flying around in close proximity to people and buildings, things can and will go wrong. Also, the advantage of having a light vehicle is that it's less likely to seriously damage itself when it falls.
I'd say some interesting design challenges when building a quadrotor are 1) Software: getting it to perform smoothly, perform stunts, and autonomous flight. 2) Weight - keeping it light and balanced 3) Reliability/Durability - build a vehicle that can crash and be recovered quickly with minimal trouble.