Mitogen Activated Protein Kinase Kinase (MEK1/2) Inhibitor Selumetinib (AZD6244 Hydrogen Sulfate) in People With Neurofibromatosis Type 1 (NF1) Mutated Gastrointestinal Stromal Tumors (GIST)

* ELIGIBILITY CRITERIA: * Age: greater than or equal to 3 years of age, BSA greater than or equal to 0.55 m\^2, and able to swallow intact capsules. * Diagnosis: must have either a clinical diagnosis of NF1 or a germline NF1 mutation, or in patients without the NF1 syndrome, demonstrate an NF1 mutation in the GIST verified in a CLIA certified laboratory. In patients without the NF1 syndrome, confirmation of the NF1 mutation in the GIST is required for enrollment. * a) For a clinical diagnosis of NF1 patients must have at least two of the diagnostic criteria for NF1 listed below * Six or more cafe-au-lait macules (greater or equal to 0.5cm in prepubertal subjects or greater than or equal to 1.5 cm in post pubertal subjects) * Freckling in axilla or groin * A neurofibroma or plexiform neurofibroma * Optic glioma * Two or more Lisch nodules * A distinctive bony lesion (dysplasia of the sphenoid bone or dysplasia or thinning of long bone cortex) * A first-degree relative with NF1 * Patients must have a histologically or cytologically confirmed measurable GIST without PDGFRA or KIT mutations. GIST may be newly diagnosed or recurrent provided that it meets criteria for progressive or metastatic disease. Metastatic disease refers to disease outside the GI tract, not simply a multifocal primary tumor. Testing performed by the Laboratory of Pathology, NCI, unless previously conducted by a CLIA/CAP external laboratory; analysis will include evaluation of 4 exons of KIT (9, 11, 13, 17) and 3 exons of PDGFRA (12, 14, 18). * Measurable Disease: * Patients must have measurable GIST as defined by RECIST v 1.1 as at least one lesion not previously irradiated, that can be accurately measured at baseline greater than or equal to 10 mm in the longest diameter (except lymph nodes which must have short axis greater than or equal to 15 mm) with computed tomography (CT) or magnetic resonance imaging (MRI) and which is suitable for accurate repeated measurements. * Progressive disease: GIST has demonstrated progression as defined by RECIST v1.1 within the past 12 months. Patients whose tumors do not meet this criterion, and have a diagnosis of NF1, may enroll on the NF1 Natural History study. * Performance Status: ECOG less than or equal to 2 (Patients greater than or equal to 16 years of age must have a Karnofsky performance level of greater than or equal to 70% (or ECOG less than or equal to 2), and children less than or equal to 16 years old must have a Lansky performance of greater than or equal to 70% * Patients must have normal organ and marrow function as defined below: * absolute neutrophil count greater than or equal to 1,000/mcL * platelets greater than or equal to 100,000/mcL * hemoglobin (Hgb) greater than or equal to 9.0 g/dL * total bilirubin \< 1.5(SqrRoot) institutional upper limit of normal * AST(SGOT)/ALT(SGPT) \< 3.0 (SqrRoot) institutional upper limit of normal * creatinine clearance or radioisotope GFR \> 60 mL/min/1.73 m\^2 by either Cockcroft- Gault formula or analysis normal serum creatinine based on age described below: * Age (years): less than or equal to 5; Maximum Serum Creatinine (mg/dL): 0.8 * Age (years): 5 or less or equal to 10; Maximum Serum Creatinine (mg/dL): 1.0 * Age (years): 10 or less than or equal to 15; Maximum Serum Creatinine (mg/dL): 1.2 * Age (years): \>15; Maximum Serum Creatinine (mg/dL): 1.5 * Prior Therapy: Patients will be eligible if tumor is metastatic, unresectable, progressive, or if complete tumor resection is not considered to be feasible without substantial risk or morbidity. * Since there is no standard effective chemotherapy for patients with NF1 and GIST, patients may be treated on this trial without having received prior medical therapy directed at their GIST. Patients who have had prior GIST-directed surgery may enroll provided they have measurable disease. * Since selumetinib is not expected to cause substantial myelosuppression, there will be no limit to number of prior myelosuppressive regimen for GIST or other tumor manifestations associated with NF1. * Patients who have received previous investigational agents or biologic therapy, such as tipifarnib, pirfenidone, Peg-Intron, sorafenib, imatinib or other targeted therapies are eligible for enrollment. At least 4 weeks must have elapsed since receiving medical therapy directed at the PN and patients who received previous GIST-directed therapy must either demonstrate progression as defined by RECIST, or be unable to tolerate their previous therapy. Patients who received effects of all prior therapy to less than or equal to grade 1 before entering this study. * Cytotoxic chemotherapy last dose must have been received at least 28 days prior to enrollment, their last dose of biological therapy, immunomodulatory agents, vaccines, differentiating agents, used to treat their cancer at least 7 days prior to enrollment, their last dose of a monoclonal antibody at least 30 days prior to enrollment, and their last dose of any investigational agent at least 30 days prior to enrollment. * Growth factors that support platelet or white cell number or function must not have been administered within the 7 days prior to enrollment. * At least 6 weeks must have elapsed prior to enrollment since the patient received any prior radiation therapy. * At least 4 weeks must have elapsed since any surgeries, with evidence of good wound healing. * The effects of selumetinib on the developing human fetus at the recommended therapeutic dose are unknown. For this reason, 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, for the duration of study participation, and for 4 weeks after dosing with selumetinib ceases. Women of child-bearing potential must have a negative pregnancy test prior to entry. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, the patient should inform her treating physician immediately. Please note that the selumetinib manufacturer recommends that adequate contraception for male patients should be used for 16 weeks post-last dose due to sperm life cycle. * All patients and/or their parents or legal guardians must sign a written informed consent. * Willingness to avoid excessive sun exposure and use adequate sunscreen protection if sun exposure is anticipated. * Willingness to avoid the ingestion of grapefruit and Seville oranges (as well as other products containing these fruits, e.g. grapefruit juice or marmalade) during the study, as these may affect selumetinib metabolism. * Although not a requirement, participants will be asked to also participate in protocol 10-C-0086 Comprehensive Omics Analysis of Pediatric Solid Tumors and Establishment of a Repository for Related Biological Studies. Patients with NF1 will be asked to coenroll on the NF1 Natural History Study and 08-C-0079: Natural History Study and Longitudinal Assessment of Children, Adolescents, and Adults with Neurofibromatosis Type 1. EXCLUSION CRITERIA: * Patients with evidence of another malignancy or benign tumor requiring chemotherapy or radiation therapy are excluded; however, those patients with a plexiform neurofibroma requiring treatment will be eligible as selumetinib has documented activity in plexiform neurofibromas. * Patients with a diagnosis of NF1 and GIST who do not meet other eligibility criteria may enroll on the NF1 Natural History Study, and will be followed on this study. Should they require therapy for GIST based on evidence of progression, they may then enroll on study. * Patients who are receiving any other investigational agents. * Prior therapy with selumetinib or another specific MEK inhibitor is not permitted. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to selumetinib or other agents used in study. * Previous MEK, RAS, or RAF inhibitor use. * Patients who anticipate the need for surgical intervention within the first three cycles (3 months), as surgical intervention during the period of DLT evaluation may affect analysis of adherence and/or make the subject inevaluable. * Patients with known brain metastases should be excluded from this clinical trial because of their poor prognosis and because they often develop progressive neurologic dysfunction that would confound the evaluation of neurologic and other adverse events. * Patients with the following cardiac conditions are excluded: * Uncontrolled hypertension (Adults: blood pressure \[BP\] of greater than or equal to 150/95 despite medical support/management. Participants 18 years of age and younger should have a blood pressure less than or equal to 95th percentile for age, height and gender. Preexisting hypertension in adults should be controlled (either with pharmacological or nonpharmacological methods) at the time of enrollment.) * Acute coronary syndrome within 6 months prior to starting treatment * Uncontrolled angina - Canadian Cardiovascular Society grade II-IV despite medical support/management * Heart failure NYHA Class II or above * Prior or current cardiomyopathy including but not limited to the following: Known hypertrophic cardiomyopathy * Known arrhythmogenic right ventricular cardiomyopathy * Baseline left ventricular ejection fraction (LVEF) less than or equal to 55% * Previous moderate or severe impairment of left ventricular systolic function (LVEF less than 50% on echocardiography or equivalent on Multi-Gated Acquisition Scan \[MUGA\]) even if full recovery has occurred. * Severe valvular heart disease * Atrial fibrillation with a ventricular rate greater than 100 bpm on ECG at rest * Ophthalmological conditions as follows: * Current or past history of retinal pigment epithelial detachment (RPED)/central serous retinopathy (CSR) or retinal vein occlusion * Intraocular pressure (IOP) greater than 21 mmHg or uncontrolled glaucoma (irrespective of IOP). * Evidence of optic glioma, malignant glioma, malignant peripheral nerve sheath tumor, * Ophthalmological findings secondary to long-standing optic pathway glioma (such as visual loss, optic nerve pallor or strabismus) or long-standing orbitotemporal PN (such as visual loss, strabismus) will NOT be considered a significant abnormality for the purposes of the study * Subjects with any other significant abnormality on ophthalmic examination (performed by an ophthalmologist) should be discussed with the Study Chair for potential eligibility. * Inability to swallow capsules, since capsules cannot be crushed or broken. * Patients with refractory nausea and vomiting, chronic gastrointestinal (GI) diseases (e.g., inflammatory bowel disease) or significant bowel resection that may significantly alter the absorption of study agent * Patients receiving any medications or substances that are strong inhibitors or inducers of CYP 1A2, 2C8, 2C9, 2C19, 3A4/5 and UGT 1A1 and 1A3, P-glycoprotein, or BCRP are ineligible. Because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference. As part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product. * No supplementation with vitamin E is permitted because the selumetinib capsules contain vitamin E * 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. * Pregnant women are excluded from this study because the effects of selumetinib on the developing human fetus at the recommended therapeutic dose are unknown. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother, breastfeeding should be discontinued if the mother is treated with selumetinib. * HIV-positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with selumetinib. Appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated. * Ongoing radiation therapy, chemotherapy, hormonal therapy, immunotherapy, or biologic therapy directed at the tumor. Those patients with a plexiform neurofibroma requiring treatment will be eligible, as selumetinib has documented activity in plexiform neurofibromas. * Patients who in the opinion of the investigator may not be able to comply with the safety monitoring requirements of the study.