High spectral efficiency and low latency are required to provide ubiquitous communication for the emerging applications in 5G wireless communication networks. In this letter, we propose a novel framework that considers these requirements simultaneously by integrating the notion of effective capacity (EC) into orthogonal frequency division multiple access (OFDMA) cloud-radio access networks (C-RAN) where the users select the distributed radio remote heads (RRHs) based on their specific delay requirements to transmit over different subcarriers cooperatively. Consequently, an optimization problem is defined to maximize the EC under the average peak power constraint and the delay requirements. The problem is combinatorial and non-convex and an algorithm based on the duality and alternating optimization algorithms is proposed, which is efficiently computed with good accuracy. Simulation and analytical results demonstrate that the proposed solution has a near-optimal performance and there is a trade-off between delay and spectral efficiency. Moreover, the cooperation between RRHs can considerably improve the system throughput.