Functional split is a promising technique to flexibly balance the processing cost at remote ends and the fronthaul rate in cloud radio access networks (C-RAN). By harvesting renewable energy, remote radio units (RRUs) can save grid power and be flexibly deployed. However, the randomness of energy arrival poses a major design challenge. To maximize the throughput under the average fronthaul rate constraint in C-RAN with renewable powered RRUs, we first study the offline problem of selecting the optimal functional split modes and the corresponding durations, jointly with the transmission power. We find that between successive energy arrivals, at most two functional split modes should be selected. Then the optimal online problem is formulated as an Markov decision process (MDP). To deal with the curse of dimensionality of solving MDP, we further analyze the special case with one instance of energy arrival and two candidate functional split modes as inspired by the offline solution, and then a heuristic online policy is proposed. Numerical results show that with flexible functional split, the throughput can be significantly improved compared with fixed functional split. Also, the proposed heuristic online policy has similar performance with the optimal online one, as validated by simulations.