Iodine I-131 With or Without Selumetinib in Treating Patients With Recurrent or Metastatic Thyroid Cancer

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…