The original title of the play was not “Oedipus Rex” but “Oedipus Tyrannus." Which title is more appropriate?
Confusion about the title arises from translation of the original Greek in which Sophocles wrote the play. Reading the information at the site listed below will definitely help you understand that the English title Oedipus Rex, which is a Latin version of the title with Rex meaning king, is not a direct translation of the Greek, in which basileus means king. The word basileus, a hereditary monarch, doesn't appear in the play. Not even Jocasta is referred to as basileia, the form for queen.
Tyrannos, on the other hand, is ambiguous, as the author of the article notes. In Greek it can mean a self-appointed ruler with a military background; today the connotations include cruelty and oppression of citizens. However, Greek has no other word for a well-intentioned ruler who rules alone. Oedipus was elected or appointed king, according to this source, which is part of The Classics Pages. Tyrannos becomes his title by default simply because the Greeks have no other word in their language.
Technically speaking, those who interpret Tyrannos as an indication of his true character do Oedipus a disservice by implying he is a tyrant in the modern sense of the word. As noted in sagetrieb's answer, nothing in the play supports this interpretation.
Oedipus does not start out as a tyrant; indeed, if he did, the play would not be a tragedy because his fall would have occurred before the action began. Nor does he become a tyrant. He has hubris to be sure, and he is driven to find the truth, but as a tyrant in doing this or in ruling his people, I find no evidence to suggest that. He seeks to know his identity, a quest indicative of human life, and he hopes that by doing this he will cleanse the state, Thebes, of its pollution. Oedipus Rex is the appropriate title.
I agree with that answer, not only because Oedipus is not a tyrant, but because hs is a true king. Rex means King. Oedipus needs to seek his identity, and suffers this conflict, because of the prophecy about him becoming king. As a king, he cares very much for his country and believes he must know his identity because a king must be an honorable figure for his people. Therefore, since all the action surrounds his position as King, Oedipus Rex is appropriate.