One of the greatest challenges in solar and stellar physics in the coming years will be to observe the second solar spectrum with a spatial resolution significantly better than 1". This type of scattering polarization observation would probably allow us to discover hitherto unknown aspects of the Sun's hidden magnetism. Here we report on some theoretical predictions for the photospheric line of Sr I at 4607 Å, which we have obtained by solving the three-dimensional (3D) radiative transfer problem of scattering line polarization in a realistic hydrodynamical model of the solar photosphere. We have taken into account not only the anisotropy of the radiation field in the 3D medium and the Hanle effect of a tangled magnetic field, but also the symmetry-breaking effects caused by the horizontal atmospheric inhomogeneities produced by the solar surface convection. Interestingly, the Q/I and U/I linear polarization signals of the emergent spectral line radiation have sizable values and fluctuations, even at the very center of the solar disk where we observe the forward-scattering case. The ensuing small-scale patterns in Q/I and U/I turn out to be sensitive to the assumed magnetic field model and are of great diagnostic value. We argue that it should be possible to observe them with the help of a 1 m telescope equipped with adaptive optics and a suitable polarimeter.