This randomized phase I trial is studying the side effects and the best dose of selumetinib when given together with paclitaxel as a second line therapy in treating patients with stage IIIB-IV non-small cell lung cancer (NSCLC). Selumetinib may stop or slow the growth of tumor cells by blocking a protein called mitogen-activated protein kinase (MEK) that is needed for cell growth. Drugs used in chemotherapy, such as paclitaxel, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving selumetinib together with paclitaxel may kill more tumor cells.