Reconfigurable intelligent surfaces (RISs) are considered as an effective means to improve both the spectral efficiency and coverage in wireless systems. By properly setting the phase shift matrix, RIS can enhance the propagation environment. In this paper, we investigate an RIS aided mobile edge computing (MEC) network in the finite blocklength (FBL) regime with the goal to maximize the energy efficiency under both coding length and maximum decoding error rate constraints. We first investigate the single user equipment (UE) scenario and propose a three-step optimization algorithm. To extend the system model, we further investigate the two-UE scenario where non-orthogonal multiple access (NOMA) transmission is adopted. A revised three-step optimization algorithm is demonstrated to address the problem. Numerical results verify that the proposed three-step optimization algorithms can solve the problems in both scenarios efficiently. In particular, the numerical results show that loosening the FBL constraint can improve the performance and a larger CPU frequency at the MEC server leads to an improved energy efficiency. It is also noted that adjusting the RIS phase shift matrix can enhance the signal-to-noise ratio (SNR)/signal-to-interference-plus-noise ratio (SINR) at the BS and improve the decoding error rate under both scenarios.