We present the galaxy stellar mass function and its evolution in clusters from z˜ 0.8 to the current epoch, based on the WIde-field Nearby Galaxy-cluster Survey (WINGS) (0.04 ≤z≤ 0.07), and the ESO Distant Cluster Survey (EDisCS) (0.4 ≤z≤ 0.8). We investigate the total mass function and find that it evolves noticeably with redshift. The shape at M* > 1011 M&sun; does not evolve, but below M*˜ 1010.8 M&sun; the mass function at high redshift is flat, while in the local Universe it flattens out at lower masses. The population of M*= 1010.2-1010.8 M&sun; galaxies must have grown significantly between z= 0.8 and 0. We analyse the mass functions of different morphological types (ellipticals, S0s and late types), and also find that each of them evolves with redshift. All types have proportionally more massive galaxies at high than at low-z, and the strongest evolution occurs among S0 galaxies. Examining the morphology-mass relation (the way the proportion of galaxies of different morphological types changes with galaxy mass), we find it strongly depends on redshift. At both redshifts, ˜40 per cent of the stellar mass is in elliptical galaxies. Another ˜43 per cent of the mass is in S0 galaxies in local clusters, while it is in late types in distant clusters. To explain the observed trends, we discuss the importance of those mechanisms that could shape the mass function. We conclude that mass growth due to star formation plays a crucial role in driving the evolution. It has to be accompanied by infall of galaxies on to clusters, and the mass distribution of infalling galaxies might be different from that of cluster galaxies. However, comparing with high-z field samples, we do not find conclusive evidence for such an environmental mass segregation. Our results suggest that star formation and infall change directly the mass function of late-type galaxies in clusters and, indirectly, that of early-type galaxies through subsequent morphological transformations.