The Hunger Games and the Greek myth of Theseus and the Minotaur show many similarities. There are many accounts of Suzanne Collins herself saying that she drew a lot of inspiration from the story. Gresh's The Hunger Games Companion outlines both the similarities and differences nicely, but I think there is a lot of room to build upon her observations.
Gresh makes a nice little chart for her readers that goes through each similarity and difference she noticed, so I think I'll go in that same order, outlining what she says, and adding my own notes as we go along.
- Theseus has an evil step-mother, bent on killing him so as to protect her royal claims. While Gresh correctly points out that Katniss does not have an "evil step-mother", she certainly has someone who wants her dead so as to to protect his lifestyle - President Snow. He does not try to poison Katniss, as Medea does Theseus, but he is definitely terrified of her and the power he knows she could have.
- More on the Medea/Snow dichotomy, Snow does not send Katniss out to kill a bull, but he does give her a task that he makes quite plain will kill her if she does not succeed. "Convince me." He says to Katniss, concerning her and Peeta's relationship. Like Theseus, if the task is not successfully completed, she will die.
- Theseus is the son of a king and/or god, and a king's daughter. Katniss is the daughter of a mere coal miner and healer. Clearly, this is a significant difference. Theseus is presumably wealthy, despite the fact that he was hidden from his father; after all, his mother is still a princess. Katniss, from District 12, is very poor, facing hunger and poverty on a daily basis. I find it interesting that in the end they both have the same desires - to end something horrific, whether it be the sacrifices or the controlling government of Panem, despite their different upbringings.
- A labyrinth is created in the myth, similar to the different arenas of the Hunger Games. Every year, in both stories, an equal amount of girls and boys are sent into this labyrinth/arena to their deaths. The labyrinth, containing the minotaur, and the tributes sent into it, are a punishment by a more powerful being (Poseidon in the myth, the government in the Hunger Games) for disobeying (again, Poseidon was angry that Minos had defied him and not sacrificed his bull; Panem was paying its people back for a previous revolution).
- Both Theseus and Katniss sacrifice themselves. Theseus had a more general reasoning for his sacrifice: hoping to end the sacrifice of children, he volunteered himself for the whole. Katniss, on the other hand - at first at least - volunteered herself solely to protect her little sister, Prim. Over the course of the books, however, she does sacrifice herself in a different way. She gives herself as the face of the rebellion in order to overthrow the government.
- Gresh did not mention this next similarity, perhaps because it is not so concrete as the others, but upon reading the myth, I thought this was a kind of interesting connection. I relate the minotaur to Panem as a whole. The minotaur is a beast created because of the displeasure of a higher power; in the case of the myth, Poseidon. In the same way, Panem is a creation of the displeasure of its own government, because of the uprising of the Districts 74 years previously. The minotaur is fed children yearly to pay for Minos' mistake. Similarly, the Capitol is - figuratively - fed children every year to pay for the Districts' mistake.
These are just some of the what I am sure are innumerable connections Suzanne Collins forged between the myth of Theseus and the Minotaur and her own work, The Hunger Games.