"Kamen Rider Fruit Ninja" is a popular one. And once the series started getting darker, "Gaim of Thrones", and it's not exactly hard as to figure how/where that nickname came from.
Ryoji and Hideyasu collectively are referred to as the "Nutty Buddies". This receives a "Funny Aneurysm" Moment when Hideyasu tells Ryoji that he was just using him. Later subverted tragically when Hideyasu is proven to be a lot less callous about Ryoji, albeit by that point being far too late
There was originally a rumor going around that the mysterious girl was named "Eve" in reference to the Biblical Forbidden Fruit. After that was debunked, the fandom has settled for calling her things like "Helheim Mai", "Alt-Mai", or "Not!Mai" due to her resemblance to regular Mai. She's eventually revealed to be Future Mai, and the end of the series makes her a metaphorical Eve even if she's never named as such.
Mitsuzane is occasionally referred to as "Mitch" as an Americanized version of his in-series nickname, Micchy.
Æsir Fansubs translated the Sengoku Driver as "Wärring Driver" in order to convey the pun behind the namenote It's written with the Kanji for "War" and "Extreme", with their dubbing the Hešvy MŽtal ‹mlaut as appropriately extreme. When it turned out that the Driver was actually named for its creator, Ryoma Sengoku, and Æsir changed his surname as well, fans took to calling him "Warren Statesman".
Another nickname relating to translations but not involving the subs this time, Takatora is sometimes called "Hawktiger" because his given name is a homophone for the Japanese words for those animals (though it actually translates as "Noble Tiger"). Of course, Kamen Rider fans would recognize those as two of the three animals which make up Kamen Rider OOO's Tatoba Combo, leading to the creation of a humorous picture of our Overseer with grasshopper legs to complete the trio. Doubles as Awesome Mc Coolname.
Alternately, some have taken to calling him "OO", since he's one Medal short of being OOO.
The "Energy Riders", for the four Yggdrasill Riders with Genesis Drivers (Zangetsu Shin, Duke, Sigurd, and Marika).
Team PopUp = "Team Sexy", thanks to the Kamen Rider Girls' portrayal.
As of the halfway mark, some have started using "Sour Grapes" as a nickname for Jerk Ass!Micchy since, true to the phrase's origins, he's acting this way out of jealousy over the attention and affection Kouta gets from Mai.
More derisive viewers call him Bitchy-Micchy.
Sid gets the nicknames like Cherry Juice, Crushed Cherries, or Cherry Crush, thanks to Roshuo squeezing him to death inside a cliff face.
The battle in #36 has been dubbed "Melonbowl 2014" because of the Riders involved Takatora as Zangetsu versus Mitsuzane as Zangetsu Shin.
Kamen Rider Ryugen's weapon is called the Grape Ryuhou note "Dragon Gun", and when he transforms, the Sengoku Driver calls out "Ryuhou, Ha Ha Ha!". Ryuhou is also, although written with different kanji note "Peak" instead of "gun", the name of a character (coincidentally Chinese) from another NichiAsa Kids Time show.
While not a straight example, you have to wonder if naming someone Takatoranote Only the kanji for 'tora' (tiger) is the correct one was deliberate.
Gaku Sano (Kouta/Gaim) and Gaku Matsuda (Zack/Knuckle).
Hideyasu Jonouchi shares part of his name with Jonouchi, whose most memorable trait outside from his hotheadedness was his bond with Yugi. Hideyasu, however, isn't hot-headed. His friend is though... or rather was before he ditched him, making Hideyasu the total opposite of Joey.
Subverted with the Gaim Gaiden movies, which introduces a second watermelon-themed arms; the one from the series is called Suika Arms, while the one from Gaiden is Watermelon Arms (and is said to be the prototype of the former).
No Export for You: South Korea skipped from Wizard to Drive. Presumably they didn't like the samurai motif.
No Stunt Double: Minami Tsukui portrays Yoko Minato/Kamen Rider Marika in both human and Rider forms and also performs her own stunts.
The Other Darrin: Kataoka Ainosuke VI does not reprise his role as Kougane in the final episode.
Gaku Sano, who plays Kouta/Gaim, wished he could be a character in Kamen Rider Kuuga when he was growing up, now he gets to be The Hero of the series.
Shōnan no Kaze, the band doing the series theme song, are mostly either fans of the series or at least its previous bands.
The actor for Kaito/Baron, Yutaka Kobayashi, was a fan of Kamen Rider Ryuki growing up. Ironically, he was not a fan of bananas, but worked to overcome this dislike after accepting his role as Kaito/Baron.
Mitsuzane/Ryugen's actor, Mahiro Takasugi, has also stated in interviews that he is a Kamen Rider Ryuki fan. Seems to be a common characteristic in the cast and crew behind Kamen Rider Gaim.
The costume for the Hekija Inves from Episode 14 is merely the Byakko Inves's, with a different mask and without the spikes in the back. Becomes a lot more meaningful once you find out the story behind these specific Inves.
Same for Episode 30's Yagi Inves, being modified from the Komori Inves's costume.
Yomotsuheguri Arms is mostly just a repainted combination of Grape's armor and Kiwi's helmet.
Kouta's outfit as The Man of the Beginning is a modified version of Kiwami Arms' silver armor with a new chestplate(which itself is a recolored Kurokage chestplate) and cape. This is justified as Kouta has now surpassed Kiwami Arms without even needing to transform.
Zangetsu's Gaiden-exclusive Watermelon Arms is simply a repainted Melon Arms wielded a repainted Melon Defender with a Gatling gun mounted on it. The Watermelon Lockseed itself is, oddly enough, a repaint of Baron's Mango Lockseed.
Similarly, Duke's Gaiden-exclusive Lemon Arms Lockseed is a repaint of Ryugen's Kiwi Lockseed.
Special mention has to go to the apple-themed Arms, which take no less than four forms over the course of the series: Mars' Golden Arms, Kamuro's Silver Arms, Jam's Darkness Arms, and Idun/Baron's Forbidden Apple Arms. Jam's case is justified, as he's actually Mars, and the Darkness Apple Lockseed is a depowered Golden Apple Lockseed.
Gaim's Daidaimaru sword is wielded by two other Riders, Bujin Gaim (where it's red rather than orange) Jam (where it's purple).
The stage shows introduce a comedy version of this with Gaim's Kabi (Moldy) Orange Arms, which is white with blue-green mold spots randomly applied.
Also from the stage show, Maja's Arms and weapon are redecos of Kamen Rider Fifteen's suit and weapons.
Real-Life Relative: Metal Yoshida, Oren's actor, and Hiromi Iwasaki, the Overlord Queen's actress, are husband and wife.
Zangetsu was supposed to be more of a rival to Gaim instead of Baron. This explains why they have the same transformation jingle and why Zangetsu was the only Armored Rider besides Kouta to have an upgrade (Kaito's Lemon Energy Arms was added later because of his popularity).
Knuckle had a faceplate designed, but since he uses a mass produced driver (which normally uses blank plates), it made no sense for it to be used.
Speaking of Knuckle, he was originally scripted to die fighting Kaito in Episode 45, but was apparently saved by Popularity Power; even Gaku Matsuda didn't know until he saw the scripts for Episode 46 and was overjoyed to find out that Zack survived.
Remember the DJ Gun combining with the Musou Saber to create Taiken Mode? There was a rumored Kiwami-exclusive combination of the DJ Gun combining with the Daidaimaru in Naginata Mode, creating the DJ Sojinto. It never materialized.
Considering how trying to wield a potential weapon like that would've been impractical due to the majority of its mass being on one side, it was probably for the best.
The biggest offender was the absence of a Jinba Melon Arms. This did appear in the toyline due to the Arms piece being able to fold into four different sides for each of the original four Energy Lockseeds, but was likely abandoned on the show due to storyline reasons. It finally appears in the Kamen Rider Gaim sequel novel, used by Takatora/Zangetsu himself.
In an interview, Takatora's actor said that there would be "very wet" bed scenes in Kamen Rider Gaim Gaiden: Kamen Rider Zangetsu. No such scene is in the final cut, likely due to content censorship. The closest we get is a Kissing Discretion Shot.
Word of God: It's said that they did a lot of Mythology Gags, such as the multiple Riders being a nod to Ryuki, or the omnipresent Yggdrasill Corporation mirroring Faiz's Smart Brain.
Zangetsu is implied to an Expy of Kamen Rider Ouja, though that could actually have been a foreshadowing of Zangetsu Shin II.
Mitsuzane (Kamen Rider Ryugen) and Takatora (Kamen Rider Zangetsu) are an Expy of Shadowmoon and Kamen Rider Black.