The expected number of vertices of $G'$ is given by a sum (over all sets of edges) that bears a certain resemblance to the Tutte polynomial $T(x,y)$ of (the graphical matroid associated to) $G$, as defined, for example, at http://en.wikipedia.org/wiki/Matroid#Tutte_polynomial. A quick calculation (maybe too quick --- I don't guarantee the following and I apologize for any errors here, but something like this should be true) indicates that, if we let $v,e,c$ be the number of vertices, edges, and components of $G$, then the expected number of vertices of $G'$ is
$$c+p^{v-c}(1-p)^{e+c-v}\frac{\partial T(x,y)}{\partial x}\left(x=\frac1p,y=\frac1{1-p}\right).$$