We present K-band data for the brightest cluster galaxies (BCGs) from the ESO Distant Cluster Survey (EDisCS). These data are combined with the photometry published by Aragon Salamanca, Baugh & Kauffmann and a low-redshift comparison sample built from the BCG catalogue of von der Linden et al. BCG luminosities are measured inside a metric circular aperture with 37 kpc diameter. In agreement with previous studies, we find that the K-band Hubble diagram for BCGs exhibits very low scatter (similar to 0.35) over a redshift range of 0 < z < 1. The colour and rest-frame K-band luminosity evolution of the BCGs are in good agreement with population synthesis models of stellar populations which formed at z > 2 and evolved passively thereafter. In contrast with some previous studies, we do not detect any significant change in the stellar mass of the BCG since z similar to 1. These results do not seem to depend on the velocity dispersion of the parent cluster. We also find that there is a correlation between the 1D velocity dispersion of the clusters (sigma(cl)) and the K-band luminosity of the BCGs ( after correcting for passive-evolution). The clusters with large velocity dispersions, and therefore masses, tend to have brighter BCGs, i.e. BCGs with larger stellar masses. This dependency, although significant, is relatively weak: the stellar mass of the BCGs changes only by similar to 70 per cent over a two order of magnitude range in cluster mass. Furthermore, this dependency does not change significantly with redshift. We have compared our observational results with the hierarchical galaxy formation and evolution model predictions of De Lucia & Blaizot. We find that the models predict colours which are in reasonable agreement with the observations because the growth in stellar mass is dominated by the accretion of old stars. However, the stellar mass in the model BCGs grows by a factor of 3-4 since z = 1, a growth rate which seems to be ruled out by the observations. The models predict a dependency between the BCG's stellar mass and the velocity dispersion (mass) of the parent cluster in the same sense as the data, but the dependency is significantly stronger than observed. However, one major difficulty in this comparison is that we have measured magnitudes inside a fixed metric aperture while the models compute total luminosities.