This paper addresses the numerical analysis of a class of a degenerate second-order evolution equations. We employ a finite element method for spatial discretization and a family of implicit finite difference schemes for time discretization. Introducing a stabilization parameter, denoted by θ, we propose a well-posed fully-discrete scheme. Sufficient conditions for its well-posedness and for quasi-optimal error estimates are established. The abstract theory is illustrated through the application to the degenerate wave equation, and numerical results validate our theoretical findings.