You asked: The exercise is to prove this representation is reducible. The hint tells me to find a common eigenvector for all 6 matrices which is just $(1,1,1)$. How do I proceed from here? Any help is appreciated.
Representation is irreducible if and only if the corresponding $FG$-module has no non-trivial $FG$-submodule. If you have managed to find a common eigenvector $v$, then you have $vg=\lambda_g v$ for each $g\in G$; which implies that $\operatorname{span}(v)$ is an $FG$-submodule.
In short: Looking for 1-dimensional submodules is the same thing as looking for common eigenvectors. If your the whole $FG$-module has dimension 3, it suffices to find out whether it has a 1-dimensional submodule, in order to decide whether it is irreducible or not.
It is perhaps worth mentioning that the same approach would work for any $S_n$ and that this representation is called permutation representation of $S_n$. Another interesting fact is that the permutation representation can be decomposed into this trivial representation and an irreducible representation of degree $n-1$. We have a question about this on this site; link to this MO thread is given there in comments.
Note: My answer is more-or-less the same as Benjalim's answer (which is deleted at the moment, so it is visible only for 10k+ users), with the exception that my answer uses modules and his answer avoids modules and uses only representations. (Both approaches, $FG$-modules and representations, are equivalent in the sense that we can get module from a representation and vice-versa. Hence we can describe properties of representation using the properties of the corresponding $FG$-module.)