My son is “Big Ham” cause he has thick, juicy legs. Also, when the wife and I talk about him, he’s referred to as “The Boy.”
Our eldest son, John, is Big Guy because of a kids’ book featuring a character with that nickname; our second son, Ethan, is Pumpkin due to his reddish-orange hair; and our youngest, Thomas, is Tommo just because he called himself that when he was very little.
I forgot one: due to our kids love of all things Star Wars, the current list of nicknames include Luke and Leia. When they’re not behaving right, those become Vader and Mara Jade. 
Pretty sickening ones, really :).
Moon Unit (which was how we referred to her before she started breathing air) also gets called “Stinkerbell”, “Baby Girl” and “Pretty Bit”. And of course a real-name version of “Dwee , uh, Moon Unit!!!”. We actually have a Christmas ornament, gifted to us by a friend when I was pregnant, labeled “Moon Unit” (said friend also gave us one with Dweezil’s real name).
Dweezil (again, a pre-breathing name) has been called Sparky, Monkey Toes, Smelly Belly and Sweetie Pete. And similarly “Moo… uh, DWEEZIL!!”.
My gf calls hers “The Old Man” (8 years old) and “Wafers” (6 years old)
I call them “Whatsisface” and “Whatsisname” Or Lemonaiden and Throckmorton.
We’ve started calling the baby “Sergeant Detritus”. I have, anyway. We also call him Chubs.
We don’t really have nicknames that we call them by but when referring to them we call them the shrimpoids, munchkins or the littlest Sir Poops-Alot.
When we are talking about them and don’t want them to pick up on it we call them ‘eldest offspring’ and ‘youngest offspring.’ They don’t know what an offspring is yet and I am not going to be the one to tell them just yet.
I call them “One” “Two” and “Three”. It started as code so my wife and I could talk about them without using their names, but once they figured it out I just started using them routinely.
Strangely enough, while I will often get their actual names mixed up, I never get 1, 2 & 3 wrong.
[Punky Monkey
We even have a song.
“Who’s my little punky monkey? Punky wunky punky monkey punky wunky monkey!”
Hey, she’s cute.](IMG_3380 (Large) | franz_laoshi | Flickr)
I have no kids, but my mom used to call me “Niffer” (short for “Jennifer”) and my father would call me “Googs” (short for “googly-goo”; nonsense baby talk).
My mom couldn’t differentiate between “things it’s ok to call your kid in private” and “things its ok to call your kid in front of her friends,” so I had to make her stop calling me Niffer when I was in high school.
My father, on the other hand, still calls me Googs every now and then (I’m 39). It’s the better of the two nicknames, IMO.
Isis Keiko is Iseiko, Icypop, Icy, isiskull, IsisCake-O-Pie, Isis-Pie-o-Cake and combinations thereof, along with lovermuffin, sweetiemuffin, sweetiemunchkin and prettymuch anything that starts with sweetie and has some sort of baked good attached, due to some non-english understanding of my Isis-Keiko turning into Isis-Cake-O somehow morphing in my GF’s mind into Isis-baked-good…
Better than Corporal Nobbs, but good heavens, why?
The VunderKind (now 23 y. o.) is still called either Bud or Rugrat.
When my daughter was about 6, she cracked an especially LOUD fart. I looked at her and told her that she tooted. Somehow, I started calling her “Toot”, which has managed to stick ever since then. She is now 29 years old.
My granddaughters name is Abby. When she was 1 or 2 (she’s 7 now) and would pitch a fit about something, it wasn’t an attitude. It was an abbytude. I still call her “The Tude”. 
I think of my 2yo niece as “la almiranta” (the admiral or the fieldmarshal) since she was about 8mo, because she’s combines natural bossiness with an amount of analytical ability I had not encountered before in little kids. She’s growing increasingly manipulative, too: I think we may have a future Presidenta in our hands, if that’s what she happens to want.
Any kids of my WoW-guildies who are shorter than the table are called “(Parent)'s gnome” by the rest of the guild, and sometimes by the parent.
Li’lbro is “Chiqui” (Li’l One), which is a traditional nickname for the youngest son (check) or a son who shares his father’s name (check). He’s also the youngest cousin, so there’s basically no way he would have been called anything else. He even gets called Chiqui by his friends sometimes. The nickname is shared by such “shorties” as Chiqui (or Txiki) Beguiristáin and Iñaki Urdangarín.
Dammit and Jesus Christ.
The Littlest Briston (a nickname that only exists here) is generally referred to as “Goose”, with the occasional “Goosie Goo” “Goober”, “Gooberini” or “Beans” thown into the mix.
My son was Punkin, which became Binkins, which became Mr. Binkins, which became Mr. B. He’s 22.
My daughter is The Pup, or She Who Must Be Obeyed. She’s 19.
They call me Dadster, so, fair trade.
Regards,
Shodan
They stopped calling my brother “sweet pea” when they learned he was diabetic.