We perform a comparative study of the spin relaxation by spin-orbit coupling induced from adatoms (hydrogen and fluorine) in graphene. Two methods are applied, giving consistent results: a full quantum transport simulation of a graphene nanoribbon, and a T-matrix calculation using Green's functions for a single adatom in graphene. For hydrogenated graphene, the dominant spin-orbit term for spin relaxation is PIA, the hitherto neglected interaction due to pseudospin inversion asymmetry. In contrast, in fluorinated graphene, PIA and Rashba couplings destructively interfere, reducing the total spin-relaxation rate. In this case we also predict a strong deviation from the expected 2:1 spin-relaxation anisotropy for out-of-plane and in-plane spin orientations. Our findings should be useful for spin-relaxation and weak-localization experiments of functionalized graphene.