Dynamical systems on curved manifolds, with a Lagrangian at most quadratic in the velocity, are considered. The classical action functional, on some finite time interval, is well defined for any smooth trial trajectory in configuration space. The action functional is at the basis of Lagrangian variational principles, from which all dynamical properties of the system can be derived. Here the problem of extending the action functional from the smooth deterministic trajectories of classical mechanics to the very irregular random trajectories of diffusions in configuration space is considered. In this way the action becomes a functional of the trial diffusion processes and can be put at the basis of stochastic variational principles. Since the problem is beset with ultraviolet divergences, the general strategy of renormalization theory is followed, by regularizing the trial diffusion processes through piecewise smooth geodesic lines for a generic given connection on the manifold. After cutoff removal and infinite counterterm subtraction, the quadratic part of the action shows a residual dependence on the generic regularizing connection field. Therefore, in the frame of this geodesic interpolation strategy, it is shown that a change in the connection field is equivalent to a well-defined renormalization of the scalar potential. These results apply to the problem of quantization of generic dynamical systems on curved manifolds, in particular to the definition of Feynman path integrals on curved configuration spaces.