I am going to use my lack of Japanese language skills and my imagination to throw some stuff together.
Aoikurenai corrected me on twitter that ガイム is pronounced [Ga-i-mu] which Google Translate told me is "Foreign affairs."
There were rumors of "Break" dealing with ghosts and/or Yokai who he puts to rest
That plus the idea of this being an armored knight or martial artist I really hope that this blends with the "disc" idea and that the name puns off of game as well so that Gaimu is
1. Kamen Rider "Game" who
2. soothes spirits (our MotW)
3. so they will be peaceful and return to the underworld and he will gain new "discs" in exchange or as thanks that
4A. will grant a modulated power infusion like the Meteor Galaxy for the four limbs + head and chest OR
4B. game discs with different "fighting styles" that he can use for different situations OR
4C. the discs can give different bits of armor on 4 limbs and head, chest and MAYBE as Lynxara, IIRC, suggested for Fourze every Nth set of discs could combine to give a new form. Maybe he would have to dance or pass something akin to a "kata trial" or whatever it is in martial arts that let's you get the next belt.