If you are interested in sticking with the egg idea, Science Olympiad used
to run an event in which the competitor had to develeop a bungee that had
an egg attached to the end. The team that came closest to hitting the
ground without breaking won. The trick was that during the design phase
engineers only knew the range of heights that the egg was going to be
dropped. They were not informed of the actual height until they arrived at
the competition. The event has since been changed so that eggs are no
longer used--both the mass and height are unknowns now!