We predict the stability of diamondoids made up of boron and nitrogen instead of carbon atoms. The results are based on quantum-mechanical calculations within density functional theory (DFT) and show some very distinct features compared to the regular carbon-based diamondoids. These features are evaluated with respect to the energetics and electronic properties of the boron nitride diamondoids as compared to the respective properties of the carbon-based diamondoids. We find that BN-diamondoids are overall more stable than their respective C-diamondoid counterparts. The electronic band-gaps (E(g)) of the former are overall lower than those for the latter nanostructures but do not show a very distinct trend with their size. Contrary to the lower C-diamondoids, the BN-diamondoids are semiconducting and show a depletion of charge on the nitrogen site. Their differences in the distribution of the molecular orbitals, compared to their carbon-based counterparts, offer additional bonding and functionalization possibilities. These tiny BN-based nanostructures could potentially be used as nanobuilding blocks complementing or substituting the C-diamondoids, based on the desired properties. An experimental realization of boron nitride diamondoids remains to show their feasibility.