Sparse subspace clustering (SSC) relies on sparse regression for accurate neighbor identification. Inspired by recent progress in compressive sensing (CS), this paper proposes a new sparse regression scheme for SSC via reweighted 1-minimization, which also generalizes a two-step 1-minimization algorithm introduced by E. J. Candès al all in [The Annals of Statistics, vol. 42, no. 2, pp. 669-699, 2014] without incurring extra complexity burden. To fully exploit the prior information conveyed by the computed sparse vector in the first step, our approach places a weight on each component of the regression vector, and solves a weighted LASSO in the second step. We discuss the impact of weighting on neighbor identification, argue that a popular weighting rule used in CS literature is not suitable for the SSC purpose, and propose a new weighting scheme for enhancing neighbor identification accuracy. Extensive simulation results are provided to validate our discussions and evidence the effectiveness of the proposed approach. Some key issues for future works are also highlighted.