The Offspring is called “Bunting” or sometimes “The Bunting” from the nursery rhyme “Bye Baby Bunting.” This evolved into Bundt Cake which sometimes turns into Cake of my Heart. Sometimes she is called Oigle or Oigle Face, which is what my mother-in-law called cute little kids.
My ex-bf and I named our first guinea pig Polpetta (RIP 1994-1998). Polpetta is Italian for meatball, and our little piggie, having black hair (they have hair, not fur), looked just like a meatball when curled up.
Sometimes I’d call her Poo-pet, because all guinea pigs do is eat and poop.
My family nickname is Beave, as in “Leave it to Beaver.” There are two stories as to the origin of that: either because I used to chew on wood (like popsicle sticks) when I was very little, or because of my buck teeth, long since corrected through orthodontia. Everyone in my family calls me that except for my mother, who calls me Sweetheart or My Sweet Baby (even though I’m 30 - I’m the baby of the family).
My nieces & nephew call me Uncle Beave, since they already have an uncle with the same first name as me. They know my real first name, but I prefer being called Uncle Beave.
A variation on that is Beave-Boy.
(God, this is embarassing.)
My sister used to call me Pêche (French for peach) when I was very young, because I had cheeks like peaches. A variation on that was Pêchie-Kins.
My sister used to call my brother Dancin’ Lad. I have no idea why, nor do I want to know. She also used to call him Squab. Again, I don’t want to know why.
I call my nephew Buddy.
My first bf called me “Goosums McGee” or “Goosie” for a while. I have no idea why. Maybe it’s because I used to goose him. 
Oh, and I call my bunny “Bunny” even though she does have a name, but my stupid best friend (who foisted Bunny on me because he went to Australia) gave her a male first name.
OK, I feel totally exposed and naked now. :eek:
My SO calls me Animal most of the time, or Jin, or some version of my name with the letters all scrambled up. He also calls me Girlfriend.
I just call him by his actual name, since that irritates him enough that I haven’t found the need for any silly-sounding pet names. Sometimes I also call him Boyfriend. He prefers “Hey you.”
My youngest son is often called Bubba because he is a chubby 8.5 month old and just looks like a bubba. Derivatives follow naturaly. Bubba Gump, Chubby bubba, bubba fett, etc.
I had a Pit Bull puppy many moon’s ago that I named Dread. My ex made me get rid of it because he used to bite our full grown Husky on the lips and just hang there. The Husky (named Kita) didn’t really care and just drug him around the house. If I ever get another Pit Bull, he will be named the same.
I have pet names for everyone. I call my parents Cat and Lemur, my SO Weasel and my friend Cheetah.
IMHO, the best name for a cat is:
Mandu
my wife calls me her “kisa” which stands for “knight im shining armor”.
Some of my friends call me “Kitten.”
My brother calls me “Cher”…not because I look like Cher, or anything…my name is Cheri. I guess he is just lazy. Hee
A former co-worker was a big baseball fan and named his dog Shortstop. I was impressed.
scott evil, your bunny should meet our bunny, who is also frequently called “Bunny”. Baby bunnies are notoriously hard to sex and this one was named Nightfall, after a female comic character, and then turned out to be a buck. Oh, well. I’m sure they’d get along famously and could commiserate over their ignorant humans.
In keeping with the thread, he is Nightfall the Wonder Bunny, King of Ghosts (after the poem by Wallace Stevens), or Evil Demon Bunny King, when his eyes glow red in the porch light.
My b/f calls me “Kitten” and “Mao-mi” (Cat, in Mandarin. He’s Caucasian; I had to teach him that name, but it was his idea originally)
I call him “Golden Bear”, a corruption of a name another friend calls him (which I also use), “Golden Boy”. He’s adopted the name “Kitsune”, too, so I also use that.
I call my red Betta (Siamese Fighting Fish) “Flammifer”, because it sounds cool, it does look like a flame, and that word was in the last line of the Lay of Earendil (AFAIK).