We don't actually know what Griever is, besides the fact that it is a GF.

Some people say it was drawn from Squall's ring, much like the GF Doomtrain, which supports in the Rinoa=Ultimecia theory.

Others say it was a legendary Gf which Squall had heard about, this keeping the ring that is supposed to represent Griever.

Neither of them has been confirmed though...