In Mexico they say that the more nicknames you have, the more you are loved. A couple that are fairly standard throughout the culture are:
gorda/o (fatty)
vieja/o (old lady/old man)
The idea is not to attract the envy and wrath of the gods, thus when someone is precious to you, you pretend that they are not.
Another we used collectively for our kids was "mocos" (snots or boogers).
My daughter, Zoe, got tired of people asking her to repeat her name and by the age of 2, when asked what her name was, would respond "Zoe, ZETA OH EH"(one hand on hip) so, of course she became "Zeta". My son, Nicolás, was a fan of the local clown TV program, "Nico Lico", and the whole family [50+ members. We never could get an accurate count]called him that. His middle name is Emiliano and we sometimes called him Nicolás Emiliano Zapata S*** P*****. He would introduce himself as such, too. As a matter of fact, in his senior yearbook, he gave that as his full name.