What does Freud mean by "the split subject"?

It probably refers to the part of the mind that the patient is unaware of: the unconscious. I'm guessing the term subject refers to the patient.

In general, neurosis can be described as any mental illness caused by distress, where the individual remains functional, such as depression, anxiety, obsessive-compulsive, low self esteem, unexplained anger, various phobias, etc. The distress is caused by unconscious conflict. Sigmund Freud coined the term "repression", where the individual suppresses instincts, such as sexuality, because of the threat of criticism or punishment from family, collegues, society, etc. Freud developed the treatment method known as psychoanalysis, which was adopted by Carl Jung and others, and is still in use today. In psychoanalysis, the doctor encourages the patient to talk about their history, life, dreams, etc. In this way, the doctor hopes to uncover what has been suppressed, and is causing the neurosis. Once discovered, and revealed to the patient, a way of properly expressing this instinct can now be searched for.