Many techniques of functional data analysis require choosing a measure of distance between functions, with the most common choice being L^2 distance. In this paper we argue that using a weighted L^2 distance, with a judiciously chosen weight function, can improve the performance of various statistical methods for functional data, including k-medoids clustering, nonparametric classification, and permutation testing. We consider three nontrivial weight functions: design density weights, inverse-variance weights, and a new weight function that minimizes the coefficient of variation of the resulting squared distance by means of an efficient iterative procedure. The benefits of weighting, in particular with the proposed weight function, are demonstrated both in simulation studies and in applications to the Berkeley growth data and a functional magnetic resonance imaging data set.