Don't Look Back: Orpheus went to The Underworld to get his wife after she died. Hades, Lord of the Underworld told him that he could take his wife so long as he didn't look back before getting out of the Underworld. He did right before leaving.
Achilles' Heel: Named for Achilles' heel. In the story of Achilles, his mother made him Nigh Invulnerable in every part of the body except for his heel. Didn't end well for him.
Achilles in His Tent: Another one named by Achilles. Achilles' Beautiful Slave Girl Briseis was taken away from him by Agammemnon, so in anger he chose not fight for the Greeks. Considering the fact that he was their ace it really screwed them over.
Cassandra Truth: Cassandra was a princess of Troy who could read the future, but was cursed to never be believed.
Launcher of a Thousand Ships: Helen of Troy is the Trope Namer. So beautiful that almost every prince in Greece wanted to marry her. Her father, who was afraid of upsetting them, was advised by Odysseus, to make them take an oath that they'd protect her should anything happen to her. Then Paris came along and eloped with her. And suddenly a 1000 Greek ships were setting off for Troy.
The Cassandra: Someone who is not believed, even though they speak the truth.
Trojan Horse: A tactic, where a group or person pretends to give up and offers a gift. They hide in that gift, in order to pass their enemies defenses and catch them off guard.
The Mentor: The guise that Athena took to advise Telemachus.
Theseus' Ship Paradox: If a ship slowly has all of it's parts replaced is it the same ship? The actual philosophy questions whether the "real ship" should be Ship A with all the original components, or Ship B with the entire history of the ship. A metaphor for human growth or something. Theseus was the guy that killed the Minotaur.