This play is an "updated" version of the Greek Tragedy Electra. Both Sophocles and Euripides wrote plays about this daughter of King Agamemnon, who was killed by his wife Clytemnestra upon returning from the Trojan War, because Agamemnon had killed their daughter Iphigenia as a sacrifice to the gods, providing the wind his ships needed to sail to Troy.
So, Lavinia and Christine are patterned after the characters Electra and Clytemnestra of these Greek Tragedies. Their relationship is one of complete lack of understanding, one for the other's situation, as was true for their Ancient Greek counterparts. Lavinia sees Christine's new love as a betrayal of her mother's relationship with her father, and she, like Electra idolizes her military hero dad. She also harbors a deep desire for revenge upon her mother for killing her father.
Between mother and daughter, there is also competition for the love of Christine's son/Lavinia's brother, Orin. This signifiesa use of the psychology of Freud and and sense of incest in this modern play that did not exist in the Ancient Greek texts, since this sort of psychological analysis of characters was not invented until the early 20th century. The terms, Oedipus Complex and Electra Complex, however, are useful in examining the familial relationships in Mourning Becomes Electra.
For more on the mother/daughter relationship in this play, the Greek Tragedies Electra and the Electra Complex, please follow the links below.