Let X be a scalar diffusion process with drift coefficient pointing towards the origin, i.e. X is mean-reverting. We denote by X* the corresponding running maximum, T-0 the first time X hits the level zero. Given an increasing and convex loss function l, we consider the following optimal stopping problem: inf(0 <=theta <= T0) E[l(X-T0* - X-theta)], over all stopping times theta with values in [0, T-0]. For the quadratic loss function and under mild conditions, we prove that an optimal stopping time exists and is defined by: theta* = T-0 boolean AND inf{t >= 0; X-t* >= gamma(X-t)}, where the boundary gamma is explicitly characterized as the concatenation of the solutions of two equations. We investigate some examples such as the Ornstein-Uhlenbeck process, the CIR-Feller process, as well as the standard and drifted Brownian motions.