Your special, fond name for your cousin is a nickname. If his name is Steven, his nickname might be Steve — or it might very well be "Stinky" or "Junior."
Nicknames can be shortened forms of first names or surnames; affectionate pet names; or names that tease or ridicule their recipient. If your name is David, but your friends call you "Dave," that's your nickname. If your sweetheart calls you "Honey," that's also a nickname. And if your coworkers call you "Skinny," there's your third nickname. Before the fifteenth century, the word was ekename, "an additional name," from the Old English eaca, "an increase."