Let $G$ be a locally compact, second countable, non-amenable group, let $X$ be a Haudorff space that is not necessarily compact, and let $G \curvearrowright X$ be a topological action that is free (i.e., all stabilizers are trivial) and minimal (i.e., every orbit is dense). I would like to show (or disprove) that if there exists a $G$-invariant mean on $X$ then there exists an $G$-invariant Borel probability measure on $X$.
This obviously holds if the action $G \curvearrowright X$ is transitive, since then it is isomorphic to the left translation action of $G$ on itself. It should also be easy to show that this is true when $G$ has property (T).
If needed, we can also assume that there exists an invariant $\sigma$-finite Borel measure on $X$, and that $X$ is "nice": Polish, locally compact, or whatever.