Cediranib Maleate With or Without Lenalidomide for the Treatment of Thyroid Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed papillary, follicular, papillary/follicular variant or Hurthle cell carcinoma; patients must be felt to be poor candidates for or refractory to further surgery or radioactive I-131 therapy; I-131 therapy must have been completed at least 4 weeks prior to enrollment; all patients are expected to be on thyroxine suppression therapy * Patients must have radiographically measurable disease; radiographically measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as \> 20 mm with conventional techniques or as \> 10 mm with spiral computed tomography (CT) scan; lesions in previously irradiated anatomic areas (external beam radiation) cannot be considered target lesions unless there has been documented growth of those lesions after radiotherapy * Patients must have evidence of disease progression (20% objective growth of existing tumors by Response Evaluation Criteria in Solid Tumors \[RECIST\] criteria) within the last 12 months * In the Phase I portion, there is no limit on prior systemic therapies (cytotoxic or targeted therapies); however, patients who have discontinued previous vascular endothelial growth factor (VEGF) inhibitors secondary to adverse events are not eligible; in the Phase 2 portion, patients cannot have received more than 1 prior chemotherapy (cytotoxic or targeted) regimen; prior VEGF-pathway inhibitors or B-RAF inhibitors are permissible * Life expectancy of greater than 12 weeks * Eastern Cooperative Oncology Group (ECOG) performance status 0-2 (Karnofsky \> 60%) * Leukocytes \> 3,000/mcL * Absolute neutrophil count (ANC) \> 1,500/mcL * Platelets \> 100,000/mcL * Hemoglobin \> 9 g/dL * Serum calcium \< 12.0 mg/dL * Total serum bilirubin below or equal to upper limit of institutional normal * Patients with hyperbilirubinemia due to Gilbert's syndrome may enroll in the trial * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal * Creatinine below or equal to upper limit of institutional limits OR creatinine clearance \> 50 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * Patients must have corrected QT interval (QTc) \< 480 msec * The following groups of patients are eligible provided that they have New York Heart Association (NYHA) class II cardiac function on baseline echocardiogram (ECHO)/multi-gated acquisition scan (MUGA): * Those with a history of class II heart failure who are asymptomatic on treatment * Those with prior anthracycline exposure * Those who have received central thoracic radiation that included the heart in the radiotherapy port * Females of childbearing potential (FCBP) must have a negative serum or urine pregnancy test with a sensitivity of at least 25 mIU/mL within 10-14 days prior to and again within 24 hours of starting lenalidomide and must either commit to continued abstinence from heterosexual intercourse or begin TWO acceptable methods of birth control, one highly effective method and one additional effective method AT THE SAME TIME, at least 28 days before she starts taking lenalidomide; FCBP must also agree to ongoing pregnancy testing; men must agree to use a latex condom during sexual contact with a FCBP even if they have had a successful vasectomy; all patients must be counseled at a minimum of every 28 days about pregnancy precautions and risks of fetal exposure * A female of childbearing potential is a sexually mature woman who: 1) has not undergone a hysterectomy or bilateral oophorectomy; or 2) has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months) * Females of childbearing potential (FCBP) who receive cediranib alone must also have a negative initial and ongoing pregnancy tests as described above; FCBP who receive cediranib alone must also commit to continued abstinence from heterosexual intercourse or begin TWO acceptable methods of birth control, one highly effective method and one additional effective method AT THE SAME TIME, at least 28 days before she starts taking cediranib; men on cediranib alone must agree to use a latex condom during sexual contact with a FCBP even if they have had a successful vasectomy; all patients receiving cediranib alone must be counseled at a minimum of every 28 days about pregnancy precautions and risks of fetal exposure * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier; at least 4 weeks must have elapsed since any major surgery; patients with prior use of thalidomide or lenalidomide are excluded * Patients may not be receiving any other investigational agents * Patients with known brain metastases should be excluded because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events; Note well (N.B): Patients with brain metastases with stable neurologic status following local therapy (surgery or radiation) for at least 8 weeks from definitive therapy and without neurologic dysfunction that would confound the evaluation of neurologic and other adverse events are eligible for participation * History of allergic reactions attributed to compounds of similar chemical or biologic composition to cediranib, lenalidomide, or other agents used in this study * Patients with poorly controlled hypertension (systolic blood pressure of 140 mmHg or higher or diastolic blood pressure of 90 mmHg or higher) are ineligible * Patients with 1+ or greater proteinuria on urinalysis should collect a 24 hour urine collection; patients with greater than 1.5 gram protein/24 hours are excluded * Because lenalidomide may increase the risk of deep vein thrombosis (DVT) or pulmonary embolism (PE), patients must stop Epogen (epoetin alfa) at least 4 weeks prior to enrollment * Patients with any condition (e.g., gastrointestinal tract disease resulting in malabsorption, prior surgical procedures affecting absorption, or active peptic ulcer disease) that impairs their ability to absorb cediranib tablets or lenalidomide capsules are excluded; however, for patients who are unable to swallow cediranib tablets, cediranib tablets may be administered as a dispersion in water (ie, either drinking water, sterile water \[for injection\] or purified water); cediranib can be administered via nasogastric tube or gastrostomy tube; for patients unable to swallow lenalidomide whole, lenalidomide can be administered via gastrostomy feeding tube * Patients with any of the following conditions are excluded: * Serious or non-healing wound, ulcer, or bone fracture * History of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within the past 28 days of treatment * Any history of cerebrovascular accident (CVA) or transient ischemic attack within 12 months prior to study entry * History of myocardial infarction, cardiac arrhythmia, stable/unstable angina, symptomatic congestive heart failure, or coronary/peripheral artery bypass graft or stenting within 12 months prior to study entry * History of pulmonary embolism within the past 12 months * Class III or IV heart failure as defined by the NYHA functional classification system * Patients with uncontrolled intercurrent illness including, but not limited to, ongoing or active infections or psychiatric illnesses/social situations that would limit compliance with study requirements are ineligible * Pregnant women are excluded from this study because cediranib and lenalidomide are agents with the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with cediranib or lenalidomide, breastfeeding should be discontinued if the mother is treated with cediranib with or without lenalidomide * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with cediranib or cediranib with lenalidomide; in addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy; appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated.