What is this sauce-making technique called? My mother was not the best cook, well she cooked because she needed to, but I don't think she ever enjoyed it. One of the things she did was if she was making sauce, is to take a spoonful of margarine or butter and use a fork to knead a little bit of flour into the fat. 

She would then slowly add milk to the butter paste to get the thickness she wanted, and then flavour it depending on what sauce she wanted to make.

Is there a name for this? Was she making a roux or a beschamel without even knowing what either is? Is this a French technique?
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.