Club Yasen - "Yasen" is Japanese for "night battle". In Kantai Collection, Naka's elder sister Sendai is obsessed with night battles, to the point where the other kanmusu have nicknamed her yasen baka ("the night battle idiot"). In the context of the game, night battle literally means continuing the battle after nightfall; in the fandom (and encouraged by a number of winky-face dialogue lines in game) it has The Obvious Other Meaning as well.

Navy Special No. 5 - Marine heavy fuel oil, also known as Bunker B. As Mark Knopfler teaches us, if you wanna run cool, you got to run on heavy fuel.

aidoru style - Canonically, Naka does consider herself an idol (in the Japanese pop singer sense); that is in fact how she introduces herself, as "the fleet's idol, Naka-chan." Here we've branched her out a bit, musically speaking.

fairies - Equipment fairies turn up a lot in KanColle. They pilot the miniature airplanes launched by carriers, and represent damage control crews and various pieces of equipment (radar sets, particular weapons, and so on) in fit-out screens. In some fan interpretations, they also operate equipment at sea (gun turrets, for example, are sometimes shown being worked by fairy guncrews).

Not all equipment in the game is represented by fairies; sometimes the weapons themselves are "personified" as sort-of-robotic sidekicks (such as the gun turrets of Amatsukaze, Shimakaze, and the Akizuki-class anti-aircraft destroyers). There seems to be no consistent pattern to this beyond what a particular ship's artist thought would be cute.

TR-808 Rhythm Composer - The iconic drum machine of the 1980s and '90s, in hip-hop and rap as well as techno and pop (which is why the 808 fairy, who can be seen on the 808, directly in front of Naka, is rocking the hip-hop style).

2-4-11 Projekt - Scrapping Naka in-game yields two units of fuel, four ammunition, and 11 steel. She's a common drop and her personality annoys some players, so the running (rather cruel) joke in fandom is that she gets scrapped a lot. The above is presumably not what the name of her act means in-story, but...

Also, here's a clearer look at the logo:

Why "Projekt"? I don't really know. It just seemed more techno when I was making the logo. :)

scrapped once before she was even launched - The Sendai class was planned to consist of eight ships, but only four were under construction when the Japanese government signed the Washington Naval Treaty of 1922, which severely limited Japan's options in regards to new construction. The IJN brass decided that all future cruiser development would be focused on heavy cruisers; the four Sendai-class ships on which construction hadn't yet started were canceled, and the two least far along were ordered scrapped on the ways. Naka, ship number three, was one of the latter, but her demolition was delayed by the 1923 Great Kanto Earthquake, after which someone reconsidered and she was completed and launched after all.

The Washington Treaty and the 1923 earthquake figure large in the histories of a number of the most famous IJN ships of World War II, and thus of a number of KanColle characters, most notably the aircraft carriers Akagi and Kaga (both of which were converted from other ships rendered superfluous during construction by the treaty, in Kaga's case because Akagi's sister ship was too badly damaged by the earthquake for her planned carrier conversion to be finished).