For instance let $f:[0,1]\to[0,1]$ be the Cantor function and define $g(x):=x+f(x)$. Then $g:[0,1]\to[0,2]$ is a homeomorphism that maps the complement of the Cantor set $C$ onto a measure one open set of $[0,2]$ (just because $g'(x)=1$ on $[0,1]\setminus C$). So $g_{|C}:C\to g(C)$ is a homeomorphism of the Cantor set onto a compact set of measure one, and if $W$ is any non-measurable subset of $g(C)$, $g$ is also a homeomorphism between the Lebesgue measurable null-set $g^{-1}(W)$ and $W$.

**edit.** As to the issue of finding a non-measurable subset within a Lebesgue measurable set of positive measure $S$, there is such a set of the form $S\cap (V+q)$, the trace on $S$ of a suitable translation of the Vitali set $V$. Indeed, the Vitali set $V$ does not contain any measurable subset of positive measure (reason: if $E\subset V$ then $E-E \subset V-V\subset \big(\mathbb{R}\setminus\mathbb{Q}\big)\cup \{0 \}$, while $E-E$ is always a nbd of zero for any measurable set of positive measure $E$). On the other hand, the sets $S\cap (V+q)$ for $q\in\mathbb{Q}$ are a countable cover of $S$, so one of them has positive exterior measure.