She had a hard time pronouncing his first name. In the book "Tripwire," it is revealed that Reacher's mother had a heavy French accent, and she had a hard time pronouncing the name "Jack." As a result, she would often call him by his last name instead.
She wanted to show him respect. In some cultures, it is considered respectful to address someone by their last name. Reacher's mother may have been trying to show her son respect by calling him by his last name.
She wanted to distinguish him from his brother. Reacher has a younger brother named Joe. His mother may have called him by his last name in order to distinguish him from his brother.
She was trying to teach him humility. Reacher's mother may have been trying to teach her son humility by calling him by his last name. She may have wanted him to understand that he was not better than anyone else, and that he should not expect to be treated any differently.
She was trying to toughen him up. Reacher's mother may have been trying to toughen her son up by calling him by his last name. She may have wanted him to understand that he would have to face challenges in life, and that he would need to be strong and resilient.
She was trying to instill a sense of duty in him. Reacher's mother may have been trying to instill a sense of duty in her son by calling him by his last name. She may have wanted him to understand that he had a responsibility to others, and that he should always be willing to help those in need.
She was trying to instill a sense of pride in him. Reacher's mother may have been trying to instill a sense of pride in her son by calling him by his last name. She may have wanted him to understand that he was part of a family with a long and proud history, and that he should be proud of his heritage.
She was trying to show him love. In some cultures, it is considered a sign of love to address someone by their last name. Reacher's mother may have been trying to show her son how much she loved him by calling him by his last name.
It was just a habit. It's possible that Reacher's mother simply called him by his last name out of habit. She may have called him that way for so long that it became second nature.