The simplest idea is to generate a sequence of points $(X_i,Y_i)$, where $X_i,Y_i$ are independent uniform random variables, in a rectangle containing the polygon (the minimal rectangle bounding the polygon). Then, you simply drop the points which fall outside the polygon. You continue until you get $n$ unrejected points.
As we could expect, this problem has been considered before by many people. You might find this one very useful. The basic question is how fast/simple you want to generate the $n$ points. The method I described is most simple, but might be time consuming.
EDIT: To add some mathematics to this post, let us justify the rejection method and triangulation approaches (though they might seem intuitively clear).
Rejection method approach. Suppose that $(X,Y)$ is uniformly distributed in a rectangle $R$ bounding the polygon $M$, and that $S$ is an arbitrary square contained in $M$. Then,
$$
{\rm P}((X,Y) \in S |(X,Y) \in M) = \frac{{{\rm P}((X,Y) \in S )}}{{{\rm P}((X,Y) \in M)}} =
\frac{{{\rm area}(S) /{\rm area}(R)}}{{{\rm area}(M)/{\rm area}(R)}} = \frac{{\rm area}(S) }{{{\rm area}(M)}}.
$$
Hence, given that $(X,Y) \in M$, $(X,Y)$ is uniformly distributed in $M$.
Triangulation approach. Suppose that the polygon $M$ is partitioned into triangles $T_1,\ldots,T_m$, and set $p_i = {\rm area} (T_i)/ {\rm area}(M)$. Suppose that ${\rm P}((X,Y) \in T_i)=p_i$ and that, given $(X,Y) \in T_i$, $(X,Y)$ is uniformly distributed in $T_i$. Now, for fixed $k$, let $S \subset M$ an arbitrary square contained in $T_k$ (think of arbitrarily small squares). Then,
$$
{\rm P}((X,Y) \in S) = \sum\limits_{i = 1}^m {{\rm P}((X,Y) \in S|(X,Y) \in T_i )p_i } = {\rm P}((X,Y) \in S|(X,Y) \in T_k )p_k.
$$
Since, given that $(X,Y) \in T_k$, $(X,Y)$ is uniformly distributed in $T_k$, we have
$$
{\rm P}((X,Y) \in S|(X,Y) \in T_k ) = \frac{{{\rm area}(S)}}{{{\rm area}(T_k )}}.
$$
Hence,
$$
{\rm P}((X,Y) \in S) = \frac{{{\rm area}(S)}}{{{\rm area}(T_k )}}\frac{{{\rm area}(T_k )}}{{{\rm area}(M)}} = \frac{{{\rm area}(S)}}{{{\rm area}(M)}}.
$$
We conclude that $(X,Y)$ is uniformly distributed in $M$.