What is this sauce-making technique called?
If she was doing this in a pan on the heat (melting the butter, stirring in the flour, then adding milk), this is called making a roux, then a béchamel. If, instead, she kneaded the flour and butter 'cold', then added this to a hot liquid, it is called beurre manié. Notice that in both cases, the sauce is heated. From your question I could not tell if, and at what point, your mother introduced heat to the butter-flour(-milk?) mixture.