The question of how much of human action is directed by free will and how much is determined by fate has fascinated people from the Greeks to the present. Think about this issue in historical terms, considering the impact of the natural sciences on this debate or in philosophical terms, researching the ideas of Existentialist writers like Jean Paul Sartre and Albert Camus.
In many ways, Electra is a powerful woman and can be seen as someone driven toward a higher purpose by her profound inner strength. How do her actions fit into the Greek definition of hero? Is that definition different for a woman that it would be for a man? You might research classical mythology generally (reading Ovid's Metamorphosis, for example) or Homer's Odyssey. Or you could compare Electra with another of Sophocles's titular heroines, Antigone.
The cycle of plays of which Electra can be seen as a part raise important issues about the relationship between divine law to human law. Try to develop an independent standard of criteria by which people might act ethically. You might look at classical writers like Aristotle (the Ethics or Politics) or you might research into the relationship between law and literature.
Critics have argued that while Sophocles's play is entitled Electra, Electra herself is not really central to the play's action. They contend that she stands around speaking while those around her act. Do you agree?