Both of them are out late, and neither has anyone to go home to.
Eleven o’clock at night is still evening, and both of them are out late because neither has anyone to go home to. When Roger tries to snatch Mrs. Jones’s purse, she realizes that he has nowhere else to go. She wants to take him home because she knows that there is no one in his home that is taking care of him. The fact that he has no one to make sure that his face is washed, and he has eaten dinner, and he has money for what he needs (or wants) makes her sympathetic.
The woman said, "You ought to be my son. I would teach you right from wrong. Least I can do right now is to wash your face. Are you hungry?"
"No’m," said the being dragged boy. "I just want you to turn me loose."
In the end, the boy decides to stay with her. (She lives alone, and notice that she has no one there either.) He has an option to leave, but chooses not to. He goes to the sink, and washes up. He even offers to go to the store for her. He wants to be trusted. She is the mother figure that he does not have. She recognizes that, and he recognizes that.
Even though it is late, the two of them still have not eaten—or at least Roger has not eaten, because, as I said, there is no one at his house.
“…Maybe, you ain’t been to your supper either, late as it be. Have you?"
"There’s nobody home at my house," said the boy.
"Then we’ll eat," said the woman, "I believe you’re hungry—or been hungry—to try to snatch my pocketbook."
Yes, eleven o’clock is late at night. However, both of them were still out and about. Neither has anyone. They find each other with this interaction, and they each give the other something needed. Mrs. Jones felt the need to mother the boy, or she would have turned him in to the police instead of taking him home. Roger felt the need to be mothered, or he would not have stayed.