She is simply called "The Mother" because there is no name referenced in the game or by the developer.
She is believed to have divorced The Father before the events of the game, and it is unknown whether she won or lost custody to her son. He is believed to be taken back into custody by his father at the end of the game.
In The Game Edit
As the game opens, she was celebrating the child's second birthday. After answering the door to an unknown caller, she hands her son a present, a teddy bear, then leaves him upstairs in his bedroom with the new toy.
Later, she puts her son to bed. He wakes in the night to find Teddy has been taken away, and that she was missing. After finding Teddy in the washing machine, her son and Teddy journey through a variety of twisted paranormal rooms with the recurring theme of disturbing hidden drawings, wine bottles, scattered across the floor and move in boxes, in order to seek out memories to help them find the boys mother. While trying to find the memories, her son was hunted by three frightening monsters in the shape of a coat, a stranger with an umbrella, and a large angry woman.
In the final area, her son begins to hear her and the father arguing over their divorce in muffled and distorted shouting tone. After the last memory was found, Teddy is captured by the monster and his arm torn off. Afterwards, her son has repeated visions of her, when it is revealed that after she was divorced, she subsequently fell into depression and severe alcoholism. The bottles seen throughout the game are a cruel reminder to her son of the thing that makes her turn into an abusive monster. The entire story up to that point had been a scared child's way of making sense of an extremely troubled childhood, and the monsters chasing him was in fact the abusive mother herself who had been tormenting her son for 2 years and the father who had been trying to protect the toddler by taking him away from her. Teddy is also revealed to be a present given to her son by the father.
Once the memories are completed, the child will return to the house, in which they will find her sobbing on the kitchen floor after having torn off Teddy's arm. When her son attempts to retrieve Teddy, she pushes him, and apologizes before resuming crying. Hearing the door knock, she will pick up Teddy and open the door. A kindly male voice will be heard, who happily greets the child and promises that he will fix Teddy.
It is speculated that the father appearing at the end was him achieving custody over her son or finally being able to take him away from her after their divorce. It is unknown when their divorce occurred, making exactly what happened slightly ambiguous. It is also unknown whether the drinking and abuse started before the divorce, or after.
In the ending of the dlc prequel right when her son was about to pick up the last toy she picks him up and looks at him very sadly. She then puts him on her shoulder and walks away to the car presumably taking him and herself to their new home away from her husband. Leaving the toy to die of freezing.