Looks like my guess was right - "LRIG" is "girl" spelled backwards, not an acronym/initialism.
This was stated by KawaseP (a.k.a. Kouhei Kawase, one of the producers for selector infected WIXOSS) in an interview with Akiba-Souken (Ctrl+F for
──LRIGs are a distinctive part of the show, aren't they?
The LRIG character designs were done by [Kyuta] Sakai, but it was Tomo Yamaguchi, who is in charge of the card game at Takara Tomy, who came up with the name "LRIG". We had a different name for them at first, but at one of the scriptwriting meetings, we decided that we wanted to go with something a bit more off-the-beaten-path, and so we decided on "GIRL" spelled backwards: "LRIG" (pronounced "loo-rig"). Up to that point, we had been making up all sorts of convoluted words, so when the simple word "LRIG" was proposed, everybody agreed.
Given the events of episode 8...
I guess we can see why the producers decided that "LRIG" was a good choice. Girls turn into LRIGs when their wishes are "fulfilled", and LRIGs likewise turn into (human) girls when that happens. Simple and effective.