An adaptive finite volume method for 2D steady Euler equations on unstructured grids is proposed. The framework of the finite volume method for the steady Euler equations follows the one in the paper [G.H. Hu, R. Li, and T. Tang, A robust WENO type finite volume solver for steady Euler equations on unstructured grids, Commun. Comput. Phys. 9 (2011) 627-648]. In this paper, we introduce the mesh adaptive methods to improve the above numerical method. The features of this work include: (i) different reconstruction stencils for WENO reconstruction are discussed in detail, including their performance on the convergence of steady state solutions and on the application of h-adaptive methods, and (ii) an effective indicator for generating quality nonuniform mesh is proposed, which is based on the entropy production. The improvement of the numerical methods is demonstrated by plenty of numerical examples.