Smallest set not included in a collection of sets
Let $[n] = \{ 1, 2, \dotsc, n \}$, and let $\mathcal{F} = \{S_1, S_2, \dotsc, S_m \} \subseteq 2^{[n]}$ be the input set family. Unless I misunderstood your problem formulation, we want to find a minimum-size set $T \subseteq [n]$ such that $T \not\subseteq S_i$ for all $i = 1, 2, \dotsc, m$.

To answer your question, note that $T \not\subseteq S_i$ if and only if $T \cap ([n] \setminus S_i) \not= \emptyset$. That is, $T$ has to intersect the complement of each $S_i$. But this means that your problem is, essentially, equivalent to the hitting set problem (consider hitting set with input $\mathcal{G} = \{ [n] \setminus S_i \  \colon \  i = 1, 2, \dotsc, m  \}$):


  Hitting Set.
  Given a set family $\mathcal{F} \subseteq 2^{[n]}$ and integer $k$, does there exists a set $T \subseteq [n]$ with $|T| \le k$ and $T \cap S \not= \emptyset$ for all $S \in \mathcal{F}$?


Hitting set is known to be NP-complete and cannot be, loosely speaking, solved faster than in time $O(2^n)$ unless the Strong Exponential-time Hypothesis fails.