• Patient must have enrolled onto APEC1621SC and must have been given a treatment assignment to molecular analysis for therapy choice (MATCH) to APEC1621E based on the presence of an actionable mutation

‣ Note: patients with BRAF V600 actionable mutations of interest (aMOIs) will be preferentially assigned to APEC1621G (vemurafenib) if that study is open and they are otherwise eligible for it

• Patients must have a body surface area \>= 0.5 m\^2 at enrollment

• Patients must have radiographically measurable disease at the time of study enrollment; patients with neuroblastoma who do not have measurable disease but have iobenguane (MIBG) positive (+) evaluable disease are eligible; measurable disease in patients with CNS involvement is defined as tumor that is measurable in two perpendicular diameters on magnetic resonance imaging (MRI) and visible on more than one slice

‣ Note: the following do not qualify as measurable disease:

• Malignant fluid collections (e.g., ascites, pleural effusions)

∙ Bone marrow infiltration except that detected by MIBG scan for neuroblastoma

∙ Lesions only detected by nuclear medicine studies (e.g., bone, gallium or positron emission tomography \[PET\] scans) except as noted for neuroblastoma

∙ Elevated tumor markers in plasma or cerebrospinal fluid (CSF)

∙ Previously radiated lesions that have not demonstrated clear progression post radiation

∙ Leptomeningeal lesions that do not meet the measurement requirements for Response Evaluation Criteria in Solid Tumors (RECIST) 1.1

• Karnofsky \>= 50% for patients \> 16 years of age and Lansky \>= 50 for patients =\< 16 years of age; Note: neurologic deficits in patients with CNS tumors must have been stable for at least 7 days prior to study enrollment; patients who are unable to walk because of paralysis, but who are up in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score

• Patients must have fully recovered from the acute toxic effects of all prior anti-cancer therapy and must meet the following minimum duration from prior anti-cancer directed therapy prior to enrollment; if after the required timeframe, the numerical eligibility criteria are met, e.g. blood count criteria, the patient is considered to have recovered adequately

‣ Cytotoxic chemotherapy or other anti-cancer agents known to be myelosuppressive: \>= 21 days after the last dose of cytotoxic or myelosuppressive chemotherapy (42 days if prior nitrosourea)

⁃ Anti-cancer agents not known to be myelosuppressive (e.g. not associated with reduced platelet or absolute neutrophil count \[ANC\] counts): \>= 7 days after the last dose of agent

⁃ Antibodies: \>= 21 days must have elapsed from infusion of last dose of antibody, and toxicity related to prior antibody therapy must be recovered to grade =\< 1

⁃ Corticosteroids: if used to modify immune adverse events related to prior therapy, \>= 14 days must have elapsed since last dose of corticosteroid

⁃ Hematopoietic growth factors: \>= 14 days after the last dose of a long-acting growth factor (e.g. pegfilgrastim) or 7 days for short-acting growth factor; for growth factors that have known adverse events occurring beyond 7 days after administration, this period must be extended beyond the time during which adverse events are known to occur; the duration of this interval must be discussed with the study chair and the study-assigned research coordinator

⁃ Interleukins, interferons and cytokines (other than hematopoietic growth factors): \>= 21 days after the completion of interleukins, interferon or cytokines (other than hematopoietic growth factors)

⁃ Stem cell Infusions (with or without total body irradiation \[TBI\]):

• Allogeneic (non-autologous) bone marrow or stem cell transplant, or any stem cell infusion including donor lymphocyte infusion (DLI) or boost infusion: \>= 84 days after infusion and no evidence of graft versus host disease (GVHD)

∙ Autologous stem cell infusion including boost infusion: \>= 42 days

⁃ Cellular therapy: \>= 42 days after the completion of any type of cellular therapy (e.g. modified T cells, natural killer \[NK\] cells, dendritic cells, etc.)

⁃ X-ray therapy (XRT)/external beam irradiation including protons: \>= 14 days after local XRT; \>= 150 days after TBI, craniospinal XRT or if radiation to \>= 50% of the pelvis; \>= 42 days if other substantial bone marrow (BM) radiation

• Note: radiation may not be delivered to measurable disease tumor site(s) being used to follow response to subprotocol treatment

⁃ Radiopharmaceutical therapy (e.g., radiolabeled antibody, iodine I 131 metaiodobenzylguanidine \[131I-MIBG\]): \>= 42 days after systemically administered radiopharmaceutical therapy

⁃ Patients must not have received prior exposure to selumetinib (AZD6244 hydrogen sulfate)

• For patients with solid tumors without known bone marrow involvement:

‣ Peripheral absolute neutrophil count (ANC) \>= 1000/mm\^3 within 7 days prior to enrollment

⁃ Platelet count \>= 100,000/mm\^3 (transfusion independent, defined as not receiving platelet transfusions for at least 7 days prior to enrollment)

• Patients with known bone marrow metastatic disease will be eligible for study provided they meet the blood counts (may receive transfusions provided they are not known to be refractory to red cell or platelet transfusions); these patients will not be evaluable for hematologic toxicity

• Creatinine clearance or radioisotope glomerular filtration rate (GFR) within 7 days prior to enrollment \>= 70 ml/min/1.73 m\^2 or a serum creatinine based on age/gender as follows:

‣ Age: 1 to \< 2 years; maximum serum creatinine (mg/dL): male 0.6; female 0.6

⁃ Age: \>= 16 years; maximum serum creatinine (mg/dL): male 1.7; female 1.4

• Bilirubin (sum of conjugated + unconjugated) =\< 1.5 x upper limit of normal (ULN) for age within 7 days prior to enrollment

• Serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 135 U/L; (for the purpose of this study, the ULN for SGPT is 45 U/L) within 7 days prior to enrollment

• Serum albumin \>= 2 g/dL within 7 days prior to enrollment

• Shortening fraction of \>= 27% by echocardiogram, or ejection fraction of \>= 50% by gated radionuclide study within 7 days prior to enrollment

• A blood pressure (BP) =\< the 95th percentile for age, height, and gender measured within 7 days prior to enrollment; please note that 3 serial blood pressures should be obtained and averaged to determine baseline BP; patients with hypertension controlled on antihypertensive medications will be allowed if otherwise eligible

• Serum triglyceride level =\< 300 mg/dL within 7 days prior to enrollment

• Serum total cholesterol level =\< 300 mg/dL within 7 days prior to enrollment

• Patients must be able to swallow intact capsules whole

• All patients and/or their parents or legally authorized representatives must sign a written informed consent; assent, when appropriate, will be obtained according to institutional guidelines