Is it a sin for me to do housework to relieve my mother, even though she implores me by God not to do that?

What your mother did by asking in the name of Allah the Almighty is impermissible. You must obey her unless she commands disobedience, as it is obligatory to fulfill the request of one who asks in the name of Allah the Almighty if their request is permissible, without prohibition or injustice. There is no sin upon you for not fulfilling her wish due to ignorance of the ruling.