Let $V_0\leq V$ be a subspace of the vector space $V$. If $annih(V_0)\leq ann(V_0)\leq V^*$ is its annihilator, then the dual of $annih(V_0)$ ann(V_0)$ is isomorphic with the quotient space $V/V_0$:
$annih(V_0) ann(V_0) \cong \frac{V}{V_0}$
This link may be useful: http://en.wikipedia.org/wiki/Dual_space#Quotient_spaces_and_annihilators
Update:
Oops, after I added my answer, the page refreshed and I saw that you edited your comment. It seems that my answer does not apply to your question...
Anyway, I see no reason to give it a different name than annihilator, even if it annihilates a subspace of $V^*$. It has the same definition.
Update 2
It may depend on the context:
In functional analysis some use the notion of pre-annihilator because it is important to distinguish $V^{**}$ from $V$. Example: Dales, Aiena - Introduction to Banach Algebras, Operators and Harmonic Analysis.
When $V^{**}\cong V$ some don't necessarily make the distinction. Example: Marsden and Ratiu - The Breadth of Symplectic and Poisson Geometry
But it may be a good practice to specify in what space a subspace is the annihilator of a given subspace: Example: In Automorphic Forms on GL(2), Jacquet and Langlands said: "If $\tilde V_2$ is the annihilator of $V_1$ in $\tilde V$ then $V_1$ is the annihilator of $\tilde V_2$ in $V$.
For non-commutative rings, ideals, semigroups, one specifies whether the annihilator is left or right. Examples: G. Gratzer - Universal Algebra Steven G. Krantz (Ed.) - Dictionary of Algebra, Arithmetic, and Trigonometry
