Barry Mazur and I have come across the question below, motivated by (but independent of) issues regarding the Leopoldt conjecture.

Suppose that $\mathbf{C}$ is the complex numbers.

Let $H$ be a finite set, and
let $S$ be a subset of $H$. The vector space $X = \mathbf{C}^{H}$
has a canonical basis consisting of the generators $[h]$ for $h \in H$. Let
$X_S$ denote the subspace generated by $[h]$ for $h \in S$.
Suppose we consider a subspace $U$ of $X$, and ask for the
dimension of $U \cap X_S$. The answer will depend on $U$, but
we expect the codimension of the intersection will *generically*
be the sum of the codimensions of each space. In other words, let
$G(X,U)$ denote the set of vector spaces $V$
inside $X$ which
are abstractly isomorphic to $U$ - that is, $\dim(V) = \dim(U)$.
Then
$$\min \{ \dim(V \cap X_S), \ | \ V \in G(X,U)\} = \min\{0,\dim(U) + |S| - |H|\}\}.$$
Indeed, $G(X,U)$ is a Grassmannian, and equality holds on an open set in $G(X,U)$.

Suppose we now assume that $H$ is a *group*. We continue to assume that $S$ is
a subset of $H$ (not necessarily a subgroup),
and that $X_S$ is the subspace generated by $[h]$ for $h \in S$.
The vector space $X$ now has extra structure --- it has a left and right action of $H$,
indeed, $X \simeq \mathbf{C}[H]$ is just the regular representation of $H$.
Suppose that $U$ is a subspace of $X$, and let us additionally assume that
$H.U = U$, that is, $U$ is a representation of $H$, and the inclusion $U
\subseteq X$ is an inclusion of left $X = \mathbf{C}[H]$-modules. We now suppose
that $G_H(X,U)$ is the set of left $X$-modules $V$ inside $X$ which are abstractly
isomorphic to $U$ as representations of $H$ --- this is contained in but generally
much smaller than the space of vector subspaces isomorphic to $U$. The question is: can one compute
$$\delta(H,S,U) := \min \{ \dim(V \cap X_S), \ | \ V \in G_H(X,U)\}.$$
That is: what is the expected dimension of this intersection *given* the structure
of $U$ as an $H$-module?

$G_H(X,U)$ can be identified with a product of Grassmannians, namely $G(v_i,u_i)$ where $V_i$ are the irreducible representations of $H$, $v_i = \dim(V_i)$, and $u_i = \dim \mathrm{Hom}_H(V_i,U)$. A lower bound for $\delta(H,S,U)$ is given by $\min\{0,\dim(U) + |S| - |H|\}$, but this is not optimal in general.

Remark: If $u_i \in \{0,v_i\}$ for all $i$, then $G_H(X,U)$ consists of a single point.

Example: Let $H = \langle \sigma \rangle$ be cyclic of order four. Let $U$ consist of the subspace generated by $[1] + [\sigma^2]$ and $[\sigma] + [\sigma^3]$. Then $G_H(X,U) = \{U\}$ is a point. If $S = \{[1],[\sigma^2]\}$ then $\dim X_S \cap U = 1$, even though $\dim(U) + |S| -|H| = 0$.

The most general question is: Can one compute $\delta(H,S,U)$ in a nice way?

A more specific question is: Can one compute $\delta(H,S,U)$ when $H$ has an element $c$ of order two and $U = X^{c = 1}$, that is, the elements of $X$ which $c$ acts by $1$ on the right. $u_i = \dim \mathrm{Hom}_H(U,V_i)$ is equal to $\dim(V_i,c = 1)$ in this case.

An even more specific question is: Can one compute $\delta(H,S,U)$ when $H = S_4$ (with representations of dimension $1$, $1$, $2$, $3$ and $3$, and $U = X^{(12) = 1}$ has corresponding multiplicities $1$, $0$, $1$, $1$, and $2$? What about $H = D_{8}$ or $D_{10}$, and $c$ is a reflection?

Example: Suppose that $U = X^{c = 1}$, and suppose that $c$ is central in $H$. This is exactly the condition on $c$ to ensure that $G_H(X,U)$ is a point. Then $$\delta(H,S,U) = \frac{1}{2} | \{ s \in S \ | \ cs \in S\}|.$$ This generalizes the previous Example, where $c = \sigma^2$.

Let me make the following remark. As I mentioned in the question, One can obtain the complete answer when $c$ is central. In this case, one obtains generic intersections that are "larger" than what one expects from the linear algebra, at least when $|G| > 2$. It follows that a similar thing happens whenever $G$ admits a quotient $G/H$ of order $> 2$ where $c$ is central. This explains why one would expect degenerate answers in dihedral groups $D_{2n}$ of order divisible by $4$. Having done $D_3 = S_3$ by hand, the next "interesting" case is $D_5$. If I understand Greg's answer correctly, the generic intersection is always as small as possible for $D_5$ and $D_7$ and any $S$. Thus the most optimistic conjecture is that this is always the case providing that $|G^{ab}| \le 2$, for example, if $G = S_4$.