And I claim here that I know the ultimate answer to this.
Squeeze two balloons together, with an egg between them. Then permanently fix them altogether by taping around the balloons.
That's it. The total weight of two balloons and the tape used is less than 4 grams. It will win you the 1st prize ALWAYS. This works because the low weight density of the balloons serve as parachute. The system weighs 4 grams always, protecting an egg drop from 'any height' .