Actually, Odin cannot be re-named; also, Odin appears at will rather than being summoned like a conventional GF. There's nothing at all to suggest that Squall called the GF's name during that battle. Seifer's "I won't go that easily" implies knowledge of Odin's power - i.e. that he generally finishes any battle with the greatest of ease.[q=PhoenixAsh]It never says GFs can be created, but it never says where they come from either. Some are stored in powerful monsters, lamps, jewellery (that requires strange unlocking techniques), and some simply wait around for someone to prove there worth. I'm not even sure it states Griever is a normal GF anyway though I can't remeber so it's not that important.[/q]GF are described in the tutorial, and aspects of their existence can be inferred from circumstances in the game. They're magical spirits, but capable of maintaining a physical form. It's clear that they're living creatures, though. Doomtrain and Diablos aren't actually "stored" in items; Diablos dwells in another dimension, but can cross into this one periodically. Doomtrain appeared all around the world, in response to the presence of certain items. See the magazine on bizarre events... in both cases, items merely serve as the 'conduit' for accessing the GF. Other GFs appear to have played a role in peoples lives in the past; Odin has a throne in the Ruins, apparently one that's just his size. He's rather huge, so it appears to be a custom-built piece of furniture. Tonberry and Cactuar seem to have arisen from monsters.
The Odin point was fair. Though if Rinoa had asked Squall what the horseman on his ring was, he may not have said "He's called Odin, I like people who ride horses."
However, he might've said, "Horsemen are known for their fearless strength in battle"; this corresponds to what he said about lions. Rinoa then might ask him the character's name, and he'd reply. Squall never merely said "I like lions".

It's just very odd that Ultimecia would create Griever, "the most powerful HG", out of the memory of Squall or whatever, but then forget absolutely everything else relevant to the GF's existence. She clearly isn't senile; she knows what she's doing, but lacks any real knowledge of Squall and co. beyond what she saw during their possession. After all, when Edea was possessed, she tried to sacrifice Rinoa. Very stupid for Ultimecia to try to kill her 'younger self.'
I don't look for the most outlandish explanation, I look for the one that best explains the quite phenomenal amount of coincidences in the game.
'Phenomenal coincidences'? Try FFVII. (SPOILER)Cloud, as a young teen, admired Sephiroth and wanted to be more like him; similar to Squall's words about Griever. In the end, Cloud had to fight and defeat the man once considered 'the greatest'. Astonishing coincidence, or dramatic event? A 'weak' character having to overcome the supreme challenge, thereby proving his strength. Or else it could just be proof that Tifa and Jenova are the same person.