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The aim of this paper is to study the stable birational type of $Hilb^n_X$, the Hilbert scheme of degree $n$ points on a surface $X$. More precisely, it addresses the question for which pairs of positive integers $(n,n')$ the variety $Hilb^n_X$ is stably birational to $Hilb^{n'}_X$, when $X$ is a surface with irregularity $q(X)=0$. After general results for such surfaces, we restrict our attention to geometrically rational surfaces, proving that there are only finitely many stable birational classes among the $Hilb^n_X$'s. As a corollary, we deduce the rationality of the motivic zeta function $\zeta(X,t)$ in $K_0(Var/k)/([\mathbb{A}^1_k])[[t]]$ over fields of characteristic zero.