I have seen two places (one magazine I don't own and a website called "Final Fantasy Compendium", if it's still on that site, I'll check later) refer to Seymour's first form (i.e. the one where he is his everyday self) as Seymour Solus, so back in the day I had a headcanon that that was his full name.

It should be said, I have not seen it stated officially, and it's not in the game, so I don't know where they got "Solus" from. The magazine also made it look like he was only Seymour Solus after Anima had been dismissed as well.