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 $\Sigma'$.