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I keep forgetting to talk about the kanji in this series.
Gaim (Gai-mu 鎧武)
Gai 鎧 - Means armor. It's also GokaiSilver's name and probably the kanji for Kamen Rider Gai from Ryuki.
Mu 武 - Means armed/warrior. Might sound familiar since Musha (Warrior) is often heard in terms of samurai. Also the second word in Kurosawa's movie Kagemusha which means Shadow Warrior.
Okay, so keeping that in mind. Armored Rider Gaim means Armored Rider Armored Warrior.
Also, Mai 舞 is the Japanese word for dance.
It must be fun to be in an anime and just have your destiny written out for you in your name.