Aims. The galactic halo likely grew over time in part by assembling smaller galaxies, the so-called building blocks. We investigate if the properties of these building blocks are reflected in the halo white dwarf (WD) population in the Solar neighborhood. Furthermore, we compute the halo WD luminosity functions (WDLFs) for four major building blocks of five cosmologically motivated stellar haloes. We compare the sum of these to the observed WDLF of the galactic halo, derived from selected halo WDs in the SuperCOSMOS Sky Survey, aiming to investigate if they match better than the WDLFs predicted by simpler models. Methods. We couple the SeBa binary population synthesis model to the Munich-Groningen semi-analytic galaxy formation model, applied to the high-resolution Aquarius dark matter simulations. Although the semi-analytic model assumes an instantaneous recycling approximation, we model the evolution of zero-age main sequence stars to WDs, taking age and metallicity variations of the population into account. To be consistent with the observed stellar halo mass density in the Solar neighborhood (ρ 0 ), we simulate the mass in WDs corresponding to this density, assuming a Chabrier initial mass function (IMF) and a binary fraction of 50%. We also normalize our WDLFs to ρ 0 . Results. Although the majority of halo stars is old and metal-poor and therefore the WDs in the different building blocks have similar properties (including present-day luminosity), we find in our models that the WDs originating from building blocks that have young and/or metal-rich stars can be distinguished from WDs that were born in other building blocks. In practice however, it will be hard to prove that these WDs really originate from different building blocks, as the variations in the halo WD population due to binary WD mergers result in similar effects. The five joined stellar halo WD populations that we modelled result in WDLFs that are very similar to each other. We find that simple models with a Kroupa or Salpeter IMF fit the observed luminosity function slightly better, since the Chabrier IMF is more top-heavy, although this result is dependent on our choice of ρ 0 .