What does the "FG" in the formal name of the Symphogear system stand for?
In the manga translator's (http://symphogear.blogspot.com) commentary, the FG is guessed to stand for Fonic Gain (it should be Phonic Gain). However, I'd speculate that it would mean Phonic Gear as it is the more elegant and title-consistent explanation.
As we know, Symphogear could mean Symphonic Gear while Symphonic itself means (etymologically) "Sounding Together/Concomittant Sound" due to the syn- prefix means "together/with" or "concommitant"
 ref: