This phase I/II trial studies the side effects and best dose of genetically modified T cells in treating patients with stage III-IV non-small cell lung cancer (NSCLC) or mesothelioma. Many types of cancer cells, including NSCLC and mesothelioma, but not most normal cells, have a protein called Wilms tumor (WT)1 on their surfaces. This study takes a type of immune cell from patients, called T cells, and modifies their genes in the laboratory so that they are programmed to find cells with WT1 and kill them. The T cells are then given back to the patient. Cyclophosphamide and aldesleukin may also stimulate the immune system to attack cancer cells. Giving cyclophosphamide and aldesleukin with laboratory-treated T cells may help the body build an immune response to kill tumor cells.