The choice of the bandwidth in the local log-periodogram regression is of crucial importance for estimation of the memory parameter of a long memory time series. Different choices may give rise to completely different estimates, which may lead to contradictory conclusions, for example about the stationarity of the series. We propose here a data driven bandwidth selection strategy that is based on minimizing a bootstrap approximation of the mean squared error and compare its performance with other existing techniques for optimal bandwidth selection in a mean squared error sense, revealing its better performance in a wider class of models. The empirical applicability of the proposed strategy is shown with two examples: the widely analyzed in a long memory context Nile river annual minimum levels and the input gas rate series of Box and Jenkins.