This paper considers the problem of adaptive state feedback controller design for stabilizing the generalized Hamiltonian systems with unstructured components. This class of models enables one to exploit the dissipative-conservative structure of generalized Hamiltonian systems for feedback control design while relaxing the burden of deriving an exact structured model representation. First, an efficient adaptation law is designed such that a correct value of parameters is estimated. Assuming that the overall system is stabilizable, and under mild assumptions on the unstructured part of the dynamics, a stabilizing adaptive control law is designed to stabilize systems to the desired steady-state. The stability of the closed-loop system is demonstrated using Lyapunov stability arguments. A numerical illustration of the proposed approach is presented to demonstrate the potential of the design method.