We consider Einstein-Maxwell-dilaton gravity with the non-minimal exponential coupling between the dilaton and the Maxwell field emerging from low energy heterotic string theory. The dilaton is endowed with a potential that originates from an electromagnetic Fayet-Iliopoulos (FI) term in $N$ $$ \mathcal{N} $$ = 2 extended supergravity in four spacetime dimensions. For the case we are interested in, this potential introduces a single parameter α. When α → 0, the static black holes (BHs) of the model are the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) solutions. When α → ∞, the BHs become the standard Reissner-Nordström (RN) solutions of electrovacuum General Relativity. The BH solutions for finite non-zero α interpolate between these two families. In this case, the dilaton potential regularizes the extremal limit of the GMGHS solution yielding a set of zero temperature BHs with a near horizon AdS 2 × S 2 geometry. We show that, in the neighborhood of these extremal solutions, there is a subset of BHs that are dynamically and thermodynamically stable, all of which have charge to mass ratio larger than unity. By dynamical stability we mean that no growing quasi-normal modes are found; thus they are stable against linear perturbations (spherical and non-spherical). Moreover, non-linear numerical evolutions lend support to their non-linear stability. By thermodynamical stability we mean the BHs are stable both in the canonical and grand-canonical ensemble. In particular, both the specific heat at constant charge and the isothermal permittivity are positive. This is not possible for RN and GMGHS BHs. We discuss the different thermodynamical phases for the BHs in this model and comment on what may allow the existence of both dynamically and thermodynamically stable BHs.