In DyNi5, the muon localizes at the 6i (1/2,0,0.1) interstitial site and below ~80 K a second muon site becomes populated, the metastable 6ring site. We determine the mean residence time of the muon in this ring site and analyze its temperature dependence with a multiphonon quantum diffusion process. The coincidence energy is measured to be Ea/kB=400 (20) K and the tunneling matrix element J=0.21 (7) meV. A comparison with GdNi5 gives insight in the muon potential well scheme in these compounds. We attribute the large diversity of muon sites in RNi5 series, where R is a rare earth element, to the relatively low density of states at the Fermi level.