Hades - Fire/Dark - Think the Balrog from the Lord of The Rings, The Three headed Dog Cerberus, or like a 'grim reaper' archetype.
Zeus - Electric/Flying - Zeus often took the form of an eagle or bull, and was the Sky God who wielded the famous Lightning bolt. Tauros/Braviary + Zapdos?
Poseidon - Water - Any kind of large fish or sea creature, really - Dolphins, especially. Could kind of conflict with Kyogre, though.
Poseidon is also the God of Earthquakes so Water/Ground would be more fitting. An octopus or squid with a trident pitch top would be pretty cool as his pet is the Kraken (depending on what archtype stories you follow)
In terms of ideas, how about a silver deer based on Artemis? (water type perhaps?) And maybe a centaur or something with a bow for Apollo (and fire type)
Also something based on the Nemean lion, perhaps a golden lion that would be steel type (since its fur is impervious to mortal weapons)
Ooh also perhaps a mons based on the Stymphalian Birds (steel/poison since they have bronze beak and metallic feathers, and have poisonous dung apparenty)
Is this to be strictly Greek or are we just adding? I mean there are a ton of greek stories we can go off of, Jason and the Argonaughts, the trials of Hercules, Icarus, Perseus, Achellis, etc. There are a ton of Demi gods and other lesser known gods as well as the Titans and the Primodials we can use as well.
Your smaller triple legends can be based off the Muses or Furies too.