Reading Mr. Underwood’s editorial about Tom Robinson, Scout concludes, “Tom had been given due process of law to the day of his death . . . but in the secret court of men’s hearts Atticus had no case.” What does she mean?

Scout meant that the inherent prejudice in men's hearts superseded any fairness that the law offered. Scout had to reflect on the whole experience to have the wisdom to say this. She was just a child at the beginning of the book.

She was saying that all of the people in court choosing Tom's fate had too much prejudice for the outcome to be fair so Atticus had an extremely small chance of winning the case. At the beginning she would have no idea of what's right and wrong. Her brain didn't have the knowledge to process and form opinions.