There's this theory some people follow, called "R=U", which means "Rinoa=Ultimecia". They believe that Ultimecia is really a future version of Rinoa. Griever, they say, is strong evidence of this.

Personally, I reckon Griever is a legendary monster from the past, like many other GFs. Squall admires lions, and Griever is a giant, winged lion spirit, so I can understand why he'd pick Griever as a "mascot" for himself. He believes Griever to be the most powerful GF, a belief that Ultimecia shares. That'd explain why she has the GF in her possession. Being the most powerful sorceress, she'd have access to the most powerful weapons.

Also, it makes the final battle more poignant, since Squall has to fight and defeat the mythical beast he believes to be the strongest.

R=U conspiracy theorists think that the GF Griever was drawn from Squall's ring (even though the same symbol's on a lot of his stuff), or that Squall himself was transformed into Griever.
Honestly...