A transverse ferromagnetic spin-1 system with a random single-ion anisotropy is considered in the framework of an Ising model. The effective field theory and decoupling approximation are applied to the derivation of the expressions of magnetizations for a honeycomb lattice. Special emphasis is placed on the critical properties of the system. New critical properties are obtained in a certain range of single-ion anisotropy, random concentration, and transverse field.We discuss in detail the influence of the random concentration and transverse field on the critical properties. Some phenomena have not been discovered in previous reports. Detailed descriptions of the phase transition and magnetization curves are presented.