Classes I wouldn't be able to say, but jobs, on the other hand...
Green Mage: HoT-focus with some buffs. Introduce the spell Mend, which would heal for the same amount of HP Cure would, but over 8-10 seconds (with a dispel protection mechanic for PvP). An AoE version of Mend, too. It should also have a full party, 1-second cast Refresh spell.
Geomancer: Introduce Healing Earth, which would work like Sanguine Rite but to recover HP and limited to a certain HP percentage. Introduce Volatile Water, which would be a weak HoT that would heal for X amount of HP if the player with Volatile Water gets hit by any damage.
Mystic/Oracle: This would be more secondary healer. Aside from being debuff-focused, it would have a spell that would do damage and heal the party for a miniscule amount proportionate to the damage dealt. Could also have spells that reduce mob attributes and buff the party in one shot.
Thaumaturge: I'd love to see this class make a return to what it was pre-revamp but without the stupid OP DoTs and kiting spells. Sacrifice and it's line of spells made sense for the class and I feel terrible to see something like that go entirely to waste. That and I'm of the opinion that BLM was shoehorned on to THM so that people wouldn't complain about THM getting nothing when jobs were introduced.