Solving large sparse linear systems is ubiquitous in science and engineering, generally requiring iterative solvers and preconditioners, as many problems cannot be solved efficiently by using direct solvers. However, the practical performance of solvers and preconditioners is sometimes beyond theoretical analysis and an optimal choice calls for intuitions from domain experts, knowledge of the hardware, as well as trial and error. In this work, we propose a new method for optimal solver-preconditioner selection using Graph Neural Networks (GNNs), as a complementary solution to laborious expert efforts. The method is based upon the graph representation of the problem and the idea of integrating node features with graph features via graph convolutions. We show that our models perform favorably well compared with traditional machine learning models investigated by the prior literature (with an improvement of 25\% in selected evaluation metrics). Implementation details and possible limitations and improvements will be discussed.