If the domain of $B$ is the same as $A$, but you only forget the interpretation of the extra language elements, then $B$ is called a reduct of $A$ to signature $\Sigma'$. But you seem also to be taking a substructure in the smaller language. I have used the term reduct substructure to refer to this relation, but I don't know if there is some other standard terminology. The relation seems to be that $B$ is a substructure of the reduct of $A$ to $\Sigma'$.

If the domain of $B$ is the same as $A$, but you only forget the interpretation of the extra language elements, then $B$ is called a reduct of $A$ to signature $\Sigma'$. But you don't merely have a reduct, since you are taking a substructure in the smaller language. Thus, what you have is that $B$ is a substructure of the reduct of $A$ to $\Sigma'$. Having needed this concept in a recent article, I used the term reduct substructure in exactly this situation, but I haven't seen this terminology elsewhere and I don't think there is an established terminology.