In evaluating some low temperature (T < 1000 K) thermal rate coefficients for inelastic rotational excitation of H-2 by H atoms, Sun and Dalgarno have found a marked sensitivity to the potential. energy surface adopted for the calculations. We have investigated the origin of the discrepancies between previous H-3 potential energy surfaces and have developed a refined surface which addresses these concerns. New quasiclassical trajectory calculations of cross sections for low energy rotational excitation are reported. The refined surface is based on 8701 ab initio energies, most newly computed for this purpose. It has the same functional form as our earlier (BKMP) surface, but since the fit of the parameters is more fully constrained than for any previous surface it is a more accurate representation. The refined surface matches the ab initio energies with an overall rms error of 0.27 mE(h) (i.e., 0.17 kcal/mol) and a maximum absolute deviation of 6.2 mE(h) (for very compact high energy equilateral triangle conformation). For "noncompact" conformations (no interatomic distance smaller than 1.15 bohr), the rms error is 0.18 mE(h) and the maximum absolute deviation is 1.7 mE(h). The refined surface is compared critically to four previous surfaces, including the DMBE surface of Varandas et al., in several respects : Legendre expansion coefficients; the interaction region for low energy rotational excitation; near the collinear saddle point; near conical intersections of the ground and first excited state surfaces; the van der Waals well; and compact geometries. We have also compared new first excited state ab initio energies for 1809 conformations with corresponding predictions from the DMBE surface.