  Cosmological simulations of galaxy formation often rely on prescriptions for
star formation and feedback that depend on halo properties such as halo mass,
central over-density, and virial temperature. In this paper we address the
convergence of individual halo properties, based on their number of particles
N, focusing in particular on the mass of halos near the resolution limit of a
simulation. While it has been established that the halo mass function is
sampled on average down to N~30 particles, we show that individual halo
properties exhibit significant scatter, and some systematic biases, as one
approaches the resolution limit. We carry out a series of cosmological
simulations using the Gadget2 and Enzo codes with N_p=64^3 to N_p=1024^3 total
particles, keeping the same large-scale structure in the simulation box. We
consider boxes from l_{box} = 8 Mpc/h to l_{box} = 512 Mpc/h to probe different
halo masses and formation redshifts. We cross-identify dark matter halos in
boxes at different resolutions and measure the scatter in their properties. The
uncertainty in the mass of single halos depends on the number of particles
(scaling approximately as N^{-1/3}), but the rarer the density peak, the more
robust its identification. The virial radius of halos is very stable and can be
measured without bias for halos with N>30. In contrast, the average density
within a sphere containing 25% of the total halo mass is severely
underestimated (by more than a factor 2) and the halo spin is moderately
overestimated for N<100. If sub-grid physics is implemented upon a cosmological
simulation, we recommend that rare halos (~3sigma peaks) be resolved with N>100
particles and common halos (~1sigma peaks) with N>400 particles to avoid
excessive numerical noise and possible systematic biases in the results.
