AnaHol-kun: The crudely-drawn dog thing on the sign of the Animal Hospital Inori's family owns.
Cure Babaa II: Miki at first (just for looking like Karen), Setsuna later on.
mktn: Miki(-tan), for obvious reasons...
And followed by mnmk (MinaMiki), which is also a Portmanteau Couple Name of Shun Minami (Soular) and Miki.
Nozomirara: Love (Cure Peach). This nickname is a portmanteau of Nozomi and Urara, both from Yes! Precure 5 as well, and it was earned because of Love's resemblance to both.
Secchin: Setsuna, naturally. A variation is "Secchan", which becomes canon later on.
SetsuPoji: Setsuna's pose while bowling in episode 15. Was already a meme before said episode was even aired.
Ojamajo Pretty Cure - The show itself, because the character design that's similar to Ojamajo Doremi. There's a rumor that staff making this anime also called it with such nickname.
Cure Casshern - Cure Moonlight, with the similar reason like above, but for Casshern Sins instead.
Cure Baro - Dark Pretty Cure, because she shares a seiyuu with ConanEdogawa. Baro is Shinichi/Conan's way of saying bakayarou (idiot).
Cure Bubbles - Cure Marine, to go with the canon Cure Blossom. Now, if there's a third, green Precure...
The third Precure is yellow, but she fits the description of "the toughest fighter", so "Cure Buttercup" is still appropriate.
Tuxedo Megane/Tuxedo Glasses and Zetsubou Sensei II - the yet unnamed mysterious boy who rescues Cure Blossom in the second episode. The former name is given for his role similar to that of Tuxedo Kamen/Mask from "Sailor Moon", and the later for his remarkable resemblence to Zetsubou.
Nozomi is "Donut" for her hairstyle. (Or was; a character was introduced in season two who is actually named King Donut, which quickly squashed any use of the name.)
Urara is "Cream Horn" for similar reasons to Nozomi. Also, "Cure Pine-Sol", because her speech sounds like a pitch for a cleaning product.
Komachi is "Retasu" after the Tokyo Mew Mew character from whom she is obviously cloned (also, "Mama", to match "Babaa").
Kurumilk - Milk/Kurumi/Milky Rose. This name is more popular in Japanese fandom.