Should a husband put his mother or his wife first?

Both have different rights that must be fulfilled: the mother is entitled to devotion and respect, the wife to maintenance (nafkah) and good treatment.
Devotion to one's parents is a great command of Allah, but it does not permit a husband to wrong his wife. In matters of maintenance, the Shafi'i school gives the wife's maintenance priority over that of relatives, because it is a right arising from marriage that does not depend on need. In matters of obedience and respect, the husband remains obliged to treat his mother well. What is required is balance, not choosing one over the other. See Rights of husband & wife.