Extra-binomial variation in logistic linear models is discussed, among others, in Collett (1991). Williams (1982) proposed a quasi-likelihood approach for handling overdispersion in logistic regression models. Suppose we observe the number of successes $y_i$ in $m_i$ trials, for $i=1,\ldots,n$, such that
$$y_i|p_i \sim \mathrm{Binomial}(m_i, p_i)$$
$$p_i \sim \mathrm{Beta}(\gamma, \delta)$$
Under this model, each of the $n$ binomials has a different probability of success $p_i$, where $p_i$ is a random draw from a beta distribution. Thus,
$$E(p_i) = \frac{\gamma}{\gamma+\delta} = \theta$$
$$Var(p_i) = \phi\theta(1-\theta)$$
Assume $\gamma > 1$ and $\delta > 1$, so that the beta density is equal to zero at both zero and one, and thus $0 < \phi \le 1/3$. From this, the unconditional mean and variance can be calculated:
$$E(y_i) = m_i\theta$$
$$Var(y_i) = m_i\theta(1-\theta)(1+(m_i-1)\phi)$$
so unless $m_i=1$ or $\phi=0$, the unconditional variance of $y_i$ is larger than binomial variance.
Identical expressions for the mean and variance of $y_i$ can be obtained if we assume that the $m_i$ counts on the i-th unit are dependent, with the same correlation $\phi$. In this case, $-1/(m_i-1) < \phi \le 1$.
The method proposed by Williams uses an iterative algorithm for estimating the dispersion parameter $\phi$ and hence the necessary weights $1/(1+\phi(m_i-1))$ (for details see Williams, 1982).