We consider the problem of computing bounded-degree lightweight plane spanners of unit disk graphs in the local distributed model of computation. We are motivated by the hypothesis that such subgraphs can provide the underlying network topology for efficient unicasting and multicasting in wireless distributed systems. We present the first local distributed algorithm that computes a bounded-degree plane lightweight spanner of a given unit disk graph. The upper bounds on the degree, the stretch factor, and the weight of the spanner, are very small. For example, our results imply a local distributed algorithm that computes a plane spanner of a given unit disk graph U, whose degree is at most 14, stretch factor at most 8.81, and weight at most 8.81 times the weight of a Euclidean Minimum Spanning Tree of V(U). We show a wider application of our techniques by giving an O(nlogn) time centralized algorithm that constructs bounded-degree plane lightweight spanners of unit disk graphs (which include Euclidean graphs), with the best upper bounds on the spanner degree, stretch factor, and weight.