The following binning rules should be added to Simone's list, which have become even more commonplace:
Given that mutual information is the sum of marginal entropies adjusted by their joint entropy, $$I(X,Y) = H(X) + H(Y) - H(X,Y) $$
The optimal binning rule for marginal entropy $H(X)$, as well as $H(Y)$, found by Hacine-Gharbi et al. (2012)
is
$$B_X = round\bigg(\frac{\xi}{6} + \frac{2}{3\xi} + \frac{1}{3} \bigg) $$
where $\xi = \big( 8 + 324N + 12 \sqrt{36N + 729N^2}\big)^{\frac{1}{3}} $
while the optimal binning rule for joint entropy $H(X,Y)$ according to Hacine-Gharbi and Ravier (2018)
is
$$B_X = B_Y = round\Bigg[ \frac{1}{\sqrt{2}} \Bigg(1 + \sqrt{1+\frac{24N}{1-\rho^2}} \Bigg)^{\frac{1}{2}} \Bigg] $$
Applying these binning rules when measuring the individual terms of $I(X,Y)=H(X)+H(Y)−H(X,Y)$, you should have an optimally binned low-bias estimator of mutual information.
Hacine-Gharbi, A., and P. Ravier (2018): “A Binning Formula of Bi-histogram
for Joint Entropy Estimation Using Mean Square Error Minimization.”
Pattern Recognition Letters, Vol. 101, pp. 21–28.
Hacine-Gharbi, A., P. Ravier, R. Harba, and T. Mohamadi (2012): “Low Bias
Histogram-Based Estimation of Mutual Information for Feature Selection.”
Pattern Recognition Letters, Vol. 33, pp. 1302–8.