Take an empty container large enough to suspend an egg inside. Cut holes in the corners for the suspension to be attached.
Use nylons, rubber bands or other springy material to suspend the egg in the middle of the container. Draw the material through the openings you made. Staple or tape the ends of the material to the container so the egg is in the middle of the space and there is not much give to the nylon or rubber band. You do not want your egg to hit the side of the container.
Place cushion material such as bubble wrap, foam or cotton around the inside of the container. Depending on the rules for your egg drop, you may be able to fill the container with cushion, or only fill part of it.
Test your egg drop device. Practice dropping it from the required height and see if the egg breaks. Add or remove tension from your suspension material; add or remove cushioning or put your egg in a larger or smaller container, depending on how it responds to the drop.
Attach a parachute made of cloth or paper with string, straws or Popsicle sticks if allowed. A parachute will slow your egg's descent, causing a softer landing.