Inclusion Criteria:

          -  Prior treatment with therapeutic dose of radioactive iodine (> 50 mCi) with evidence
             of RAI uptake on delayed scan and with progression (biochemical or anatomic) within 12
             months of RAI

          -  Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 (Karnofsky ≥ 80%)

          -  Leukocytes ≥ 3,000/µL

          -  Absolute neutrophil count ≥ 1,500/µL

          -  Platelets ≥ 100,000/µL

          -  Total bilirubin within normal institutional limits

          -  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 within normal institutional limits OR

          -  Creatinine clearance ≥ 60 mL/min/1.73 m² for patients with creatinine levels above
             institutional normal

          -  Confirmed diagnosis of differentiated thyroid cancer (follicular or papillary thyroid
             cancer and their variants)

          -  Ability and willingness to use appropriate contraception; women of child-bearing
             potential and men must agree to use adequate contraception (hormonal or barrier method
             of birth control; abstinence) prior to study entry and for the duration of study
             participation; should a woman become pregnant or suspect she is pregnant while she or
             her partner is participating in this study, she should inform her treating physician
             immediately; men treated or enrolled on this protocol must also agree to use adequate
             contraception prior to the study, for the duration of study participation, and for 2
             weeks after completion of lenvatinib administration

          -  Patients must have measurable disease, defined as at least one lesion that can be
             accurately measured in at least one dimension (longest diameter to be recorded for
             non-nodal lesions and short axis for nodal lesions) as ≥ 20 mm (≥ 2 cm) by chest x-ray
             or as ≥ 10 mm (≥ 1 cm) with computed tomography (CT) scan, magnetic resonance imaging
             (MRI), or calipers by clinical exam

          -  Ability to understand and the willingness to sign a written informed consent document

        Exclusion Criteria:

          -  Patients who have received RAI within 12 weeks of planned retreatment

          -  Prior receipt of cumulative RAI doses in excess of 1000 mCi

          -  Patients who have not recovered from adverse events due to prior anti-cancer therapy
             (i.e., have residual toxicities > grade 1)

          -  Patients who are receiving any other investigational agents

          -  Patients with previously untreated and or symptomatic brain metastases are excluded
             from this clinical trial because of the risk of intracranial bleeding with angiogenic
             agents and tumoral swelling from RAI

          -  History of allergic reactions attributed to compounds of similar chemical or biologic
             composition to lenvatinib

          -  Uncontrolled intercurrent illness including, but not limited to, ongoing or active
             infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac
             arrhythmia, or psychiatric illness/social situations that would limit compliance with
             study requirements

          -  Patients with uncontrolled hypertension (requirement for more than 2 blood pressure
             [BP] medications or grade 2 or higher BP elevation while on adequate doses of not more
             than 2 antihypertensive agents) are excluded from the study because one of the
             significant adverse events of lenvatinib is worsening hypertension

          -  Fridericia's corrected QT (QTcF) interval prolongation greater than 500 ms

          -  Recent arterial thromboembolic event within the previous 6 months

          -  Urine dipstick proteinuria ≥ 2+ or nephrotic range proteinuria on ≥ 2 gram in 24-hour
             urine

          -  History of gastrointestinal perforation, abscess or fistula

          -  History of and or medical condition (e.g. diverticular disease; aneurysm) that
             predisposes to risk of major hemorrhage

          -  Pregnant women are excluded from this study because lenvatinib is a tyrosine kinase
             inhibitor agent 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 lenvatinib, breastfeeding should be discontinued if
             the mother is treated with lenvatinib

          -  Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral
             therapy are ineligible because of the potential for pharmacokinetic interactions with
             lenvatinib