Cooperative bit-interleaved coded modulation (BICM) is a key technology for the next-generation wireless communication systems. This paper investigates power allocation for the cooperative BICM systems with decode-remap-and-forward (DRF) relaying. Unlike the conventional decode-and-forward relays, the DRF relays may choose different constellation mappings from that of source so as to obtain a remapping gain. In spite of its importance, the power allocation in such a system has not yet been explored in the literature. Two new power allocation methods are proposed, aiming to minimize bit error rate at the destination. The first uses a cost function based on the minimum weighted squared-Euclidean distance (called PA-MWSED) and is optimized with the sub-gradient method. The second is based on a generalized MWSED (called PA-GMWSED) and is optimized with the Simplex method after the optimization is re-cast as a linear programming problem. Generally speaking, PA-MWSED has a better performance than PA-GMWSED but requires a higher complexity. Numerical results show that both of the proposed methods outperform the equal gain power allocation with large margins.