Oddly enough only a chosen few of my budgies and lovebirds have nicknames.
I'll start with the budgies: Mona, she was named after Mona Lisa due to her placidness and overall facial expression that reminds me a lot of the famous painting. My mother sometimes does call her Mona Lisa. She also goes by the name Monacita.
Luigi, got the "Lulu" nickname. He was the one who came up with the word during one of his word talking sessions, I liked it so much that sometimes I do call him Lulu.
Leonel is often called "pequenino" portuguese word for "little fellow". It was also self entitled just like Luigi.
Now for the lovebirds, I always call my boy Nico "Nikito" because he is the smallest lovebird on the flock.
Lotas is also known as Lotto.
Khaleesi only has a couple of nicknames, "Khalee" and "Bebé" (bebé is the Portuguese word for baby).
The pet that breaks all the records as far as nicknames go is my dog Elvis.
I gather he must have been called close to 100 different petnames over the 12 years I have had him and the funny thing is that he responds to them!
He has been called "Elis", I don't know why, but it took my grandmother a while to call Elvis by his rightful name and it became kind of an inside joke.
Sometimes I call him "Elvius", I like to think that would be his old Latin name. Meelo is also a favourite.
Lately I've been calling him Pinto and all different variations, Pipo (pronounced Peepo) and Pipas (pronounced Peepash).