Ender Dragons and Eggs after the 1st one, be any of the 16 game colors.

The first Ender Dragon and Dragon Egg should always spawn as black, but every Ender Dragon after that should spawn a different random dye color and drop the same colored Egg. (There would be a 1/16 chance to spawn any given color including black again.) It would be impossible to dye any Dragon Egg a different color, so whichever color it spawns as, that's what it's always going to be.

On a multiplayer server there'll be players going after the Ender Dragon over and over again. Many players will want to collect all 16 egg colors, and have two or more of the same color, and be willing to trade eggs to make their collection complete.

This would be an incentive for players on MP servers to keep on respawning and killing the Ender Dragon beyond the first player to do it who gets the egg (and if a given egg is ever lost, it can be replaced given enough time).