Mmh, let's see.
Death Knight:
Quest Name: Path of Fordragon
Other Quest Details: You'd interact with Alexstrasza and possibly acquire new information regarding the current Lich King and maybe the quest could drop hints regarding his future.
Reward: You'd gain the Fordragon's Fire/Heart of Fire or (insert name here). It would make you look more like the current Lich King. Your Frost spells would be turned into Fire spells, your eyes would glow a fiery red and well, you would look a bit more "fiery".
Druid:
Quest Name: The Celestials' Chosen
Other Quest Details: Could make the player perform tasks in Pandaria, possibly even interact with the August Celestials.
Reward: You'd gain Celestial versions of your forms.
Mage
Quest Name: Antonidas' Teachings/ (insert name here)
Other Quest Details: One idea could be that either Jaina Proudmoore or someone else (Horde players would interact with another important mage) asks you to recover some tomes that were lost during the siege of Dalaran. The player would be asked to go back in time during the siege and recover these tomes before they are destroyed and lost forever. (This would work pretty well for Alliance players. I'm not sure if it would fit the opposite faction though)
Reward: Unique casting animations. They could simply be animations that include your weapon. Or as someone else has suggested the Brilliance Aura effect (like in WC3), although I'd like to see this in the form of a glyph.
Rogue
Quest Name: Art of Deceit/Agent of Death/A Rogue's Creed
Other Quest Details: As someone else has said, the rogue would either interact with SI:7 and then visit Ravenholdt. There the player could receive contracts and the last stage would require the rogue to track down and assassinate a powerful figure, maybe in a place filled with traps and other nasty devices.
Reward: I really don't know. Maybe poison dripping from weapons, as others have suggested. Maybe the rogue could become more dark and his abilities could have a more "shadowy" look. Or maybe a unique item set.
Warrior
Quest Name: Honour/Blood & Steel
Other Quest Details:
Reward: Race specific cosmetic changes. Orc Warriors would get banners (Like the Blademasters in WC3), Dwarves could get unique dwarven weapons (like the ones Muradin uses in WC3) or maybe a unique Battle Cry (would make them shout, "For Khaz Modan!", and trigger an animation), etc.
You've all come up with nice ideas. Keep it up! I might add more later.