Inclusion Criteria:

- Patients must have histologically or cytologically confirmed papillary, follicular,
papillary/follicular variant or Hürthle 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 (PS) 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 times upper limit of normal

- Creatinine below or equal to upper limit of institutional normal 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 radiotherapy 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; 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 maleate, 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 proteinuria 1+ or greater 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 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

- 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