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.
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