  We study the origin of the diffuse stellar component (DSC) in 117 galaxy
clusters extracted from a cosmological hydrodynamical simulation. We identify
all galaxies present in the simulated clusters at 17 output redshifts, and then
build the family trees for all the z=0 cluster galaxies. Then for each diffuse
star particle identified at z=0, we look for the galaxy to which it once
belonged at an earlier redshift, thus linking the presence of the diffuse
stellar component to the galaxy formation history. The main results of our
analysis are: (i) On average, half of the DSC star particles comes from
galaxies associated with the family tree of the most massive galaxy (BCG), one
quarter comes from the family trees of other massive galaxies, and the
remaining quarter from dissolved galaxies. I.e., the formation of the DSC is
parallel to the build-up of the BCG and other massive galaxies. (ii) Most DSC
star particles become unbound during mergers in the formation history of the
BGCs and of other massive galaxies, independent of cluster mass. Our results
suggest that the tidal stripping mechanism is responsible only for a minor
fraction of the DSC. (iii) At cluster radii larger than 250 kpc/h, the DSC
fraction from the BCG is reduced and the largest contribution comes from the
other massive galaxies; in the cluster outskirts, galaxies of all masses
contribute to the DSC. (iv) The DSC does not have a preferred redshift of
formation: however, most DSC stars are unbound at z<1. (v) The amount of DSC
stars at z=0 does not corre late strongly with the global dynamical history of
clusters, and increases weakly with cluster mass. [abridged]
