We study the minimization of the positive principal eigenvalue associated to a weighted Neumann problem settled in a bounded smooth domain \Omega\subset R^N, within a suitable class of sign-changing weights. This problem naturally arises in population dynamics. Denoting with u the optimal eigenfunction and with D its super-level set associated to the optimal weight, we perform the analysis of the singular limit of the optimal eigenvalue as the measure of D tends to zero. We show that, when the measure of D is sufficiently small, u has a unique local maximum point lying on the boundary of \Omega and D is connected. Furthermore, the boundary of D intersects the boundary of the box \Omega, and more precisely, ${\mathcal H}^{N-1}(\partial D \cap \partial \Omega)\ge C|D|^{(N-1)/N} $ for some universal constant C>0. Though widely expected, these properties are still unknown if the measure of D is arbitrary. This is a joint project with B. Pellacci and G. Verzini.