Cruz is Spanish and Portuguese for “cross.” As a surname it usually marked someone who lived near a wayside cross or crossroads, or who came from one of the many places named (La) Cruz. Latin crux sits underneath the Romance word, so the meaning is not speculative.
Longer forms such as De la Cruz or Da Cruz make the “of the cross” idea explicit; Cruz can be a shortened everyday form. It is a place-type surname (feature in the landscape or a place-name), not a parent’s given name. For bilingual families, the English gloss is simply “cross,” with the reminder that the cross in question was often a landmark, not only a metaphor.