Naming humans is easy. Pick a name, that's their name.
But when there's fifty chicks named pearl running around then things get difficult. 

So for homeworld gems they are named after their gem, and may carry a nickname, but while threading with other gems of the same name they are called by a number that they are recognized as. four digit numbers only, and in the who plays who thread you can see who already has what number. 

Gems on earth may take on a human name entierly.