What is the relationship between the Oedipus complex and the character Oedipus? Is there any? It is a big jump between someone who answered a riddle to a psychological complex.
The full story (in short) of Oedipus is that his father heard a prophecy that his son will kill him. So he threw him away (basically ordered him to be killed but the killers spared him). After growing up he came back (solved the riddle on his way back) and then rebelled against his dad (without knowing that he was his father), killed him, and married his mother having children with her. After he found out that he in fact married his mother and killed his father he took his eyes out.
Basically the prophecy that Oedipus' father heard was in fact a self fulfilling prophecy. If he never had ordered his son to be killed, he would never return to kill him.
So since Oedipus killed his father, fell in love with and married his own mother, the Oedipus syndrome was named after him. To simplify things enormously, the complex refers to a subconscious love for one's mother, and thus jealousy towards one's father (as the father has a legitimate relationship with the mother).