Graph Neural Networks (GNNs) are becoming a promising technique in various domains due to their excellent capabilities in modeling non-Euclidean data. Although a spectrum of accelerators has been proposed to accelerate the inference of GNNs, our analysis demonstrates that the latency and energy consumption induced by DRAM access still significantly impedes the improvement of performance and energy efficiency. To address this issue, we propose a Memory-Efficient GNN Accelerator (MEGA) through algorithm and hardware co-design in this work. Specifically, at the algorithm level, through an in-depth analysis of the node property, we observe that the data-independent quantization in previous works is not optimal in terms of accuracy and memory efficiency. This motivates us to propose the Degree-Aware mixed-precision quantization method, in which a proper bitwidth is learned and allocated to a node according to its in-degree to compress GNNs as much as possible while maintaining accuracy. At the hardware level, we employ a heterogeneous architecture design in which the aggregation and combination phases are implemented separately with different dataflows. In order to boost the performance and energy efficiency, we also present an Adaptive-Package format to alleviate the storage overhead caused by the fine-grained bitwidth and diverse sparsity, and a Condense-Edge scheduling method to enhance the data locality and further alleviate the access irregularity induced by the extremely sparse adjacency matrix in the graph. We implement our MEGA accelerator in a 28nm technology node. Extensive experiments demonstrate that MEGA can achieve an average speedup of 38.3x, 7.1x, 4.0x, 3.6x and 47.6x, 7.2x, 5.4x, 4.5x energy savings over four state-of-the-art GNN accelerators, HyGCN, GCNAX, GROW, and SGCN, respectively, while retaining task accuracy.