In Oedipus Rex, what is the significance of Oedipus's slow coming into awareness of his identity?
You have made an excellent point. Part of the suspense of this amazing play is thanks to the way that Sophocles casts Oedipus as a kind of detective, trying to work out who killed the old King of Thebes, Laius. As the play progresses, Oedipus discovers more and more clues that continue to point towards the fact that it was he who was the murderer.
Of course, central to this process of gradual realisation is dramatic irony. Dramatic irony occurs when the reader or audience knows something important that a character in a story or drama does not know. The fundamental irony in this play stems from the fact that the audience knows King Oedipus's true identity, while he is unaware of it. Sophocles ingeniously creates numerous variations on this central irony throughout the play, increasing the audience's suspense and heightening their pity and horror.
This after all is what explains the reason why Sophocles chose to reveal the truth about the identity of Oedipus gradually - we, the audience, already know who he is, and it is with mounting terror and trepidation that we await for the inevitable to occur. Sophocles, by spinning the play out, greatly increases the suspense as we await that final moment of tremendous yet terrifying self-knowledge when Oedipus realises who he is and utters:
Oedipus, damned in his birth, in his marriage damned,
Damned in the blood he shed with his own hand!