Medical images are very sensitive information that should be well guarded, particularly in the existing healthcare systems, where information is usually shared over open networks. Medical devices that are resource-constrained require lightweight encryption methods, but the security of lightweight encryption relies on the quality of the initialization vectors (IVs). In this paper, a lightweight medical image encryption algorithm, called RabbitCheb, has been proposed that combines the Rabbit stream cipher with one-dimensional Chebyshev chaotic maps to improve IV randomness. In the suggested solution, Chebyshev chaotic sequences are used to create initialization vectors that are highly unpredictable, which enhances the randomness of the keystream generation process and does not increase the computational cost. RabbitCheb performance is tested using two medical image datasets, the Brain MRI Tumor dataset and the HAM10000 dataset. Several evaluation metrics are adopted, including entropy, NPCR, UACI, PSNR, and MSE. Experimental results demonstrate that the proposed method achieves high encryption performance, with an average entropy of 7.86, NPCR of 99.67, and UACI of 33.09, while maintaining low computational overhead. Compared with the original Rabbit cipher, RabbitCheb improves randomness and resistance against statistical and differential attacks. By employing chaos-based strategies to increase the randomness of IVs while preserving computational efficiency, this work enhances the security of lightweight stream ciphers. As a result, the proposed method is well-suited for securing medical image transmission in modern healthcare systems.