This partially randomized phase I/II trial studies the side effects and best dose of cediranib maleate when given together with or without lenalidomide and to see how well they work in treating patients with thyroid cancer. Cediranib maleate may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Lenalidomide may stop the growth of thyroid cancer by blocking blood flow to the tumor. It is not yet known whether cediranib maleate is more effective when given together with lenalidomide in treating thyroid cancer.