We investigate the properties of magnon edge states in a ferromagnetic honeycomb lattice with armchair boundaries. In contrast with fermionic graphene, we find novel edge states due to the missing bonds along the boundary sites. After introducing an external on-site potential at the outermost sites we find that the energy spectra of the edge states are tunable. Additionally, when a non-trivial gap is induced, we find that some of the edge states are topologically protected and also tunable. Our results may explain the origin of the novel edge states recently observed in photonic lattices. We also discuss the behavior of these edge states for further experimental confirmations. •Novel edge states in a ferromagnetic honeycomb lattice with armchair boundaries are obtained.•The missing bonds along the boundary sites make the edge itself a defect and Tamm-like edge states are thus induced.•The Tamm-like edge states are strongly sensitive to external on-site potentials.•The external on-site potential can also be used to tune the topologically protected edge states.