Damn, I didn't know about the Character Atk thing, thought that was Nutaku being wierd on translation as usual. My entire friendlist is full of Hecatons and Archangels.
So overall the desireable Eidolons would be:
Thunder: Kirin, Thunderbird, Phoenix (light), Ouroboros (dark)
Fire: Belial, Fafnir, Apocalypse (dark), Echidna (dark)
Water: Reiki, Frost, Fafnir (fire), Garuda (wind)
Wind: Garuda
Light: Thunderbird (thunder), Nicholas, Phoenix
Dark: Apocalypse, Echidna, Amphisbaena, Ouroboros
Honorable mentions to Takemikazuchi and Behemoth. Did I miss anyone? Ofc. situationally (especially with ele advantage) some character atk eidos might be pretty equal to better.