Typical betweenness centrality metrics neglect the potential contribution of nodes that are near but not exactly on shortest paths. The idea of this paper is to give more value to these nodes. We propose a weighted betweenness centrality, a novel metric that assigns weights to nodes based on the stretch of the paths they intermediate against the shortest paths. We compare the proposed metric with the traditional and the distance-scaled betweenness metrics using four different network datasets. Results show that the weighted betweenness centrality pinpoints and promotes nodes that are underestimated by typical metrics, which can help to avoid network disconnections and better exploit multipath protocols.