In this work, a novel and simple phase-field-based lattice Boltzmann (LB) method is proposed for the axisymmetric two-phase electro-hydrodynamic flows. The present LB method is composed of three LB models, which are used to solve the axisymmetric Allen-Cahn equation for the phase field, the axisymmetric Poisson equation for the electric potential, and the axisymmetric Navier-Stokes equations for the flow field. Compared with the previous LB models for the axisymmetric Poisson equation, which can be viewed as the solvers to the convection-diffusion equation, the present model is a genuine solver to the axisymmetric Poisson equation. To test the capacity of the LB method, the deformation of a single leaky or perfect dielectric drop under a uniform electric field is considered, and the effects of electric strength, conductivity ratio, and permittivity ratio are investigated in detail. It is found that the present numerical results are in good agreement with some available theoretical, numerical and/or experimental data.