Does anything become binding upon a young man if he breaks his promise to his mother to quit smoking, which he swore to by God?

Smoking and trading in it are forbidden, and one must abstain from it, whether you made a covenant with Allah regarding it or not. This is because your Islam is a covenant between you and Allah to perform obligations and abandon prohibitions. Your abstinence from it is further emphasized as an act of obedience to your mother. If you made a covenant with Allah in the form of an oath and then smoked again, you must offer expiation for breaking an oath.