Summary: We study the minimization of the positive principal eigenvalue associated to a weighted Neumann problem settled in a bounded smooth domain \(\Omega \subset \mathbb{R}^N, N\geq 2\), within a suitable class of sign-changing weights. This problem arises in the study of the persistence of a species in population dynamics. Denoting with \(u\) the optimal eigenfunction and with \(D\) its superlevel 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, \(\mathcal{H}^{N-1}(\partial D \cap \partial \Omega)\geq C|D|^{(N-1)/N}\) for some universal constant \(C> 0\). Though widely expected, these properties are still unknown if the measure of \(D\) is arbitrary.