The impatient and intolerant mother made every effort to_______

The mother is described as impatient and intolerant in the sentence provided. So, when they say that the mother made every effort to do something, we expect the action to reflect controlling or overbearing behaviour.

Denominate (to name or designate) is therefore rejected. Dominion is also dismissed as it is a noun. 

To domineer means to assert one's will over others in an arrogant or overbearing way, which fits the context of an impatient and intolerant mother. To dominate means to have power and influence over someone/something.  Dominate can be neutral or even positive depending on the context, whereas domineer almost always carries a negative connotation. Domineer is the better choice in this case.

The correct answer is therefore Option (A): domineer her children.