Reconfigurable intelligent surface (RIS) has emerged as a promising technology for achieving high spectrum and energy efficiency in future wireless communication networks. In this paper, we investigate an RIS-aided single-cell multi-user mobile edge computing (MEC) system where an RIS is deployed to support the communication between a base station (BS) equipped with MEC servers and multiple single-antenna users. To utilize the scarce frequency resource efficiently, we assume that users communicate with BS based on a non-orthogonal multiple access (NOMA) protocol. Each user has a computation task which can be computed locally or partially/fully offloaded to the BS. We aim to minimize the sum energy consumption of all users by jointly optimizing the passive phase shifters, the size of transmission data, transmission rate, power control, transmission time and the decoding order. Since the resulting problem is non-convex, we use the block coordinate descent method to alternately optimize two separated subproblems. More specifically, we use the dual method to tackle a subproblem with given phase shift and obtain the closed-form solution; and then we utilize penalty method to solve another subproblem for given power control. Moreover, in order to demonstrate the effectiveness of our proposed algorithm, we propose three benchmark schemes: the time-division multiple access (TDMA)-MEC scheme, the full local computing scheme and the full offloading scheme. We use an alternating 1-D search method and the dual method that can solve the TDMA-based transmission problem well. Numerical results demonstrate that the proposed scheme can increase the energy efficiency and achieve significant performance gains over the three benchmark schemes.