Inclusion Criteria:

          -  Diagnosis of recurrent and/or metastatic thyroid cancer

          -  Histological or cytological confirmation of thyroid carcinoma of follicular origin
             (including papillary, follicular, or poorly differentiated subtypes and their
             respective variants); NOTE: medullary and anaplastic thyroid cancers are excluded;
             Hurthle cell carcinomas are excluded (defined as having an invasive tumor composed of
             > 75% oncocytic [Hurthle] cells lacking the nuclear features of papillary carcinoma,
             tumor necrosis, and marked mitotic activity); patients with oncocytic (Hurthle cell)
             variants of papillary thyroid carcinoma (defined as a tumor composed of a majority of
             oncocytic [Hurthle] cells having the nuclear features of papillary carcinoma) are
             eligible to participate

          -  RAI-avid lesion on a radioiodine scan (a diagnostic, post-therapy, or post-ablation
             scans) performed =< 24 months prior to registration, which suggests that therapy with
             131I is justifiable in the judgment of the investigator

          -  Clinically or radiographically evident structural disease; patients with measurable
             disease and those with only non-measurable ("non-target") structural disease
             (according to modified Response Evaluation Criteria in Solid Tumors [RECIST] version
             [v] 1.1 criteria) are eligible;

        NOTE 1: Modification of the RECIST v1.1 measurable disease criteria includes a change in
        the definition of what is considered a measurable malignant lymph node; a malignant lymph
        node is considered measurable if any of the following apply:

          -  It is noted to be RAI-avid on radioactive iodine imaging (diagnostic or post-therapy
             whole body scans acceptable) and it measures >= 1 cm in the long axis,

          -  It is pathologically proven to be involved with thyroid cancer (by cytology or
             pathology) and it measures >= 1 cm in the long axis, or

          -  Its short axis is >= 1.5 cm when assessed by computed tomography (CT) scan NOTE 2:
             Patients only with biochemical evidence of disease without structural evidence of
             cancer are not eligible for this study

               -  For patients with non-measurable, structural disease the following must apply:

          -  Undetectable thyroglobulin antibody AND

          -  A serum thyroglobulin of 10 ng/ml or greater in the context of suppressed
             thyroid-stimulating hormone (TSH) (TSH =< 0.4 mcU/ml) =< 28 days prior to study
             registration; use of any thyroglobulin assay is allowed, though all serum
             thyroglobulin measurements for study purposes must be conducted with the same
             thyroglobulin assay

               -  Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1 or 2

               -  Able to swallow and retain orally-administered medication with no clinically
                  significant gastrointestinal abnormalities that may alter absorption

               -  Absolute neutrophil count (ANC) >= 1500/mm^3 (obtained =< 28 days prior to
                  randomization)

               -  Platelet count >= 100,000/mm^3 (obtained =< 28 days prior to randomization)

               -  Hemoglobin > 9.0 g/dL (obtained =< 28 days prior to randomization)

               -  Total bilirubin =< 1.5 x upper limit of normal (ULN) (obtained =< 28 days prior
                  to randomization)

               -  Aspartate transaminase (AST) =< 2.5 x ULN (or =< 5 x ULN in presence of liver
                  metastases) (obtained =< 28 days prior to randomization)

               -  Creatinine =< 1.5 mg/dL OR calculated creatinine clearance of >= 50 ml/min by
                  either the Cockcroft-Gault formula or 24-hours urine collection analysis
                  (obtained =< 28 days prior to randomization)

               -  Negative pregnancy test performed =< 7 days prior to registration for women of
                  childbearing potential only

               -  Provide informed written consent

               -  Willing to return to enrolling institution for follow-up (during the Active
                  Monitoring Phase of the study)

          -  Note: during the Active Monitoring Phase of a study (i.e., active treatment and
             observation), participants must be willing to return to the consenting institution for
             follow-up

               -  Willing to provide mandatory archival tumor tissue (block or minimum of 30
                  unstained slides from a primary or recurrent/metastatic thyroid cancer) for
                  correlative research purposes; NOTE: patients with less archival tumor tissue
                  available may still be eligible for the study after discussion with Academic and
                  Community Cancer Research United (ACCRU); receipt of archival tumor tissue is not
                  required for study registration and initiation of therapy

               -  Willing to provide mandatory blood samples for correlative research purposes

        Exclusion Criteria:

          -  131I therapy =< 6 months prior to registration; Note: 131I administered solely for
             diagnostic purposes is not considered 131I therapy

          -  External beam radiation therapy =< 28 days prior to registration; note: previous
             treatment with radiation is allowed if the investigator judges it will not compromise
             patient safety on the study

          -  Having been treated with a total cumulative (lifetime) 131I therapeutic activity > 800
             mCi (excluding 131I activity administered for diagnostic scans)

          -  Treatment with chemotherapy or targeted therapy (e.g. tyrosine kinase inhibitor) =< 28
             days prior to registration

          -  Prior exposure to mitogen-activated protein kinase kinase (MEK), RAS, or RAF
             inhibitors (note: previous exposure to sorafenib is allowed) OR history of
             hypersensitivity to selumetinib, thyrotropin alpha (Thyrogen), or any excipient agents

          -  Unresolved toxicity > Common Terminology Criteria for Adverse Events (CTCAE) grade 2
             from previous anti-cancer therapy, except for alopecia

          -  Cardiac conditions as follows:

               -  Uncontrolled hypertension (blood pressure [BP] >=150/95 mmHg despite medical
                  therapy)

               -  Left ventricular ejection fraction < 55% measured by echocardiography

               -  Atrial fibrillation with a ventricular rate > 100 beats per minute (bpm) on
                  electrocardiogram (ECG) at rest

               -  Symptomatic heart failure (New York Heart Association [NYHA] grade II-IV)

               -  Prior or current cardiomyopathy

               -  Severe valvular heart disease

               -  Uncontrolled angina (Canadian Cardiovascular Society grade II-IV despite medical
                  therapy)

               -  Acute coronary syndrome =< 6 months prior to registration

          -  Ophthalmological conditions as follows:

               -  Intra-ocular pressure > 21 mmHg, or uncontrolled glaucoma (irrespective of
                  intra-ocular pressure)

               -  Current or past history of central serous retinopathy or retinal vein occlusion

          -  Symptomatic or untreated leptomeningeal disease, brain metastasis, or spinal cord
             compression

          -  Unable to follow a low iodine diet or requiring medication with high content in iodide
             (e.g., amiodarone)

          -  Received iodinated intravenous contrast within =< 2 months of registration; avoidance
             of iodinated oral contrast is also preferred but not strictly required for study
             enrollment; NOTE: those who have had iodinated intravenous contrast within this time
             frame may still be eligible if a urinary iodine analysis reveals that excess iodine
             has been cleared (defined as urinary iodine documented to be < 300 mcg/day by either a
             spot urinary iodine or 24-hour urinary iodine measurement)

          -  Any of the following:

               -  Pregnant women

               -  Nursing women

               -  Men or women of childbearing potential who are unwilling to employ adequate
                  contraception

          -  Co-morbid systemic illnesses or other severe concurrent disease which, in the judgment
             of the investigator, would make the patient inappropriate for entry into this study or
             interfere significantly with the proper assessment of safety and toxicity of the
             prescribed regimens

          -  Immunocompromised patients and patients known to be human immunodeficiency virus (HIV)
             positive and currently receiving antiretroviral therapy; NOTE: patients known to be
             HIV positive, but without clinical evidence of an immunocompromised state, are
             eligible for this trial

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

          -  Receiving any other investigational agent which would be considered as a treatment for
             the primary neoplasm. (NOTE: Performance of investigational 124I PET/CT scans is
             allowed prior to and during conduct of this study)

          -  Other active malignancy =< 2 years prior to registration that will interfere with
             conduct of this trial; EXCEPTIONS: non-melanotic skin cancer or carcinoma-in-situ of
             the cervix; NOTE: if there is a history of prior malignancy, patients must not be
             receiving other specific treatment (chemotherapy, hormonal therapy, radiation) for
             their cancer

          -  Not willing to discontinue use of supplemental vitamin E