What makes us consider Oedipus a tragic hero in Oedipus Rex?
Oedipus was a tragic figure in a very counter-intuitive way. In the Greek mind only those with extraordinary characteristics can actually be tragic. Common people, in other words, cannot be tragic figures. The reason for this is because only people who transgress important boundaries through pride or what the Greeks call "hubris" can actually be tragic. The dynamic behind this is the story of the fall of a great person.
This was an important concept to the Greek. Let me unpack this a bit by looking at the oracle of Delphi.
The Greeks believed that the oracle of Delphi was the center of the world, a place of wisdom. At Delphi, there was two important slogans. The first one was, "nothing in excess." The second one was, "know yourself." Both of these slogans speak of boundaries. If you don't do anything in excess, then you are keeping within boundaries. And if you know yourself, then you will know that you are a mortal and not a god!
In the light of this, we can say that Oedipus broke boundaries. He thought he could solve any problem, such as the plague. He thought that he could outwit the gods, just like he did the sphynx. If you read the play, he drives the action completely. He leads to the investigation. He leads to his demise. Yet, on the other hand, this is tragic, because he is also so able. This is the Greek tension.