Lattice dynamical stability is an important issue for the two-dimensional (2D) monolayer materials in practical applications. In this study, the dynamical stability of 2D monolayer honeycomb materials (silicene, germanene, graphene, h-SiC, and h-BN) and the corresponding chemically functionalized structures are systematically studied using first-principles calculations within the framework of density functional theory. We calculated the phonon spectra of all materials under different strains, including zero strain, compressive strains, and tensile strains. The calculated phonon spectra show that all of the pristine and functionalized structures under zero strain are dynamically stable. With the application of strain, all of the 2D monolayer materials remain dynamically stable under the tensile strain and become dynamically instable under the compressive strains due to the observed soft modes. Using phonon eigenvector analysis and supercell optimization, it is also determined that compressive strain results in the structural fluctuations of planar monolayer materials. Despite the existence of soft modes, the chemical functionalization suppresses the dynamical instability against the compressive strains, which enhances the practicability of 2D monolayer materials. Our study provides the fundamental understanding of the mechanism of the lattice dynamical instability under the compressive strain and explores the approach for enhancing the dynamical stability of 2D monolayer materials.