Fan Nickname: Lyrical Nanoha aka: Magical Girl Lyrical Nanoha
Gaogaigar-tan - Subaru Nakajima of StrikerS, due to the blatant King of Braves references in her transformation sequence and attacks. There is a video of her transformation to the Gaogaigar transformation music here.
Due to her rather serious personality, the professional way she deals with most combat situations, and of course, her Eyepatch, some fans have referred to Cinque as "Little Boss" as a Shout-Out to Big Boss of the Metal Gear series.
The White Devil - Takamachi Nanoha herself; sweet, friendly, and very firm believer in vicious asskickings for the sake of love and justice. So named after a scene where an enemy calls her a devil, and she agrees that she is if that is what works. This is combined with her wearing all white, her proficiency at blowing things up that would make a Humongous Mecha proud, and the fact that the original Gundam was nicknamed the White Devil in Mobile Suit Gundam.
We also have the "White Devil Incident", where one of her students suffers from a Heroic BSOD and attacks her for real during training. Looking very disappointed, she catches said student's Hard Light blade, drawing blood and incidentally physically supporting her in midair, then blasts her into submission without even using her Device. People started using the nickname seriously after this.
Also known as The Demon Cannon Girl or Magical Cannon Girl, a pun on the name of her genre "mahou shoujo" (Magical Girl): "mahou" (magic/magical arts) and "ma" (demon) "hou" (cannon) are pronounced the same.
Similarly, her Catch Phrase "Zenryoku zenkai!" ("Full power!") can be also interpreted to mean "Total destruction!"
Some fans have been known to refer to her as "Sailor Beamspam".
Over on the Japanese side of the Internet, there's a growing movement to nickname her "Nanorymer" or simply "Hades" (meiou) on account of certain parallels between her fan-depiction and the title robot of Hades Project Zeorymer. This may be partly on account of how Zeorymer in the Super Robot Wars games is considered a Game Breaker.
Cardiche, aka the Fatemobile - Fate Testarossa-Harlaown's black convertible in Magical Girl Lyrical Nanoha StrikerS, So named because its black coloration and yellow hubcaps gives it a more than passing resemblance to Fate's Intelligent Device Bardiche. Unusual, because Fate's powers already allow her to travel quickly and because it wasn't an actual Testarossa.
Ditto Hückebikes, the Cool Bikes that the members of the Hückebein family ride in Force.
"Befriending" - Yes, the trope of Defeat Means Friendship has been given a nickname, due to it being the primary way Nanoha makes her friends. Used in a sentence; "She befriended the crap out of them."
The Energizer Bunny - the fluffy bunny that appears to be (or more accurately, contains) Vivio's Device in the new ViVid manga, because the cartridge system that many other Devices are equipped with uses cartridges that look a lot like batteries.
The Doom Bunny - Vita
Zaffy - Zafira. Used in the sound stages by Vivio.
Etnagito - Agito, for her resemblance to Etna of Disgaea.
Emoha - The movie manga version (which is a different continity from the movie itself) of Nanoha has this nickname. Due to her emo scream in the first chapter. The fifth chapter gives her a good reason to be emo.
With regards to the White Devil Incident, there is also "Signum Punch", or Signum's response to Teana still not getting it. (Compare Falcon Punch; coincidentally, Signum's ultimate attack from A's is Sturmfalkennote Storm Falcons.)
Reinforce Eins: A way of referring to the first Reinforce, as opposed to Reinforce "Rein" Zwei. Becomes eventually a canon name.
The Materials didn't really had official names in Battle of Aces, so fans came up with Seikou for Material-S, Rai or Raikou or Raijin for Material-L and Yami for Material-D. Until Gears of Destiny came out.
Signum Mk. II - Micaiah Chevelle, another swords(wo)man.
Chibi Ix for Ixpellia, after her "awakening" in chapter 63.