Temozolomide or Selumetinib in Treating Patients With Metastatic Melanoma of the Eye

Inclusion Criteria: * Patients must have metastatic histologically or cytologically confirmed uveal melanoma; if histologic or cytologic confirmation of the primary is not available, confirmation of the primary diagnosis of uveal melanoma by the treating investigator can be clinically obtained, as per standard practice for uveal melanoma; pathologic confirmation of diagnosis will be performed at Memorial Sloan-Kettering Cancer Center (MSKCC) or at a participating site * 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) as \>= 20 mm with conventional techniques or as \>= 10 mm with spiral computed tomography (CT) scan * Life expectancy of greater than 3 months * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 * Leukocytes \>= 3,000/mcL * Absolute neutrophil count (ANC) \>= 1,500/mcL * Platelets \>= 100,000/mcL * Hemoglobin \>= 9.0 g/dL (not requiring transfusions within the past 2 weeks) * Total bilirubin =\< 1.5 times upper limit of normal; note: patients with hyperbilirubinemia clinically consistent with an inherited disorder of bilirubin metabolism (e.g., Gilbert syndrome) will be eligible at the discretion of the treating physician and/or the principal investigator * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 times upper limit of normal for patients with no concurrent liver metastases * AST(SGOT)/ALT(SGPT) =\< 5 X institutional ULN for patients with concurrent liver metastases * Creatinine =\< 1.5 mg/dL * Tumor Gnaq and Gna11 status must be determined on all patients using a Clinical Laboratory Improvement Act (CLIA) approved assay; if initial CLIA testing is performed locally, patients must consent to provide a tumor block or unstained slides to MSKCC for central review of Gnaq and Gna11 status; this central review may be performed retrospectively and will not delay patient treatment on study * Patients must agree to provide all imaging studies for central radiology review; this central radiology review may be performed retrospectively and will not be utilized for decision making for patients on study * Ability to understand and the willingness to sign a written informed consent document * Eligibility for enrollment in each cohort is dependent upon tumor Gnaq/Gna11 status and prior therapy as follows: * Cohort 1: no prior TMZ or DTIC; mutant Gnaq/Gna11 status * Cohort 2: no prior TMZ or DTIC; wild-type Gnaq/Gna11 status * Cohort 3: received prior TMZ or DTIC; mutant or wild-type Gnaq/Gna11 status * Every effort must be made to avoid administration of drugs that are inhibitors or inducers of cytochrome P450 1A2 (CYP1A2) and CYP3A4 Exclusion Criteria: * Patients may have had any number of prior therapies, but cannot have previously been treated with a mitogen-activated protein kinase kinase (MEK) inhibitor; at least 3 weeks must have elapsed since the last dose of systemic therapy; at least 6 weeks must have elapsed if the last regimen included carmustine (BCNU), mitomycin C or an anti-CTLA4 antibody; patients must have experienced disease progression on their prior therapy in the opinion of the treating investigator * Patients may not be receiving any other investigational agents * Patients with active or untreated brain metastases; treated brain metastases must have been stable for at least 2 months * History of allergic reactions attributed to compounds of similar chemical or biologic composition to TMZ or DTIC or AZD6244 * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection or bleeding, symptomatic congestive heart failure, unstable angina pectoris, unstable cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study; breast-feeding should be discontinued if the mother is treated with AZD6244 * 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; 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 participating in this study, she should inform her treating physician immediately; please note that the AZD6244 manufacturer recommends that adequate contraception for male patients should be used for 16 weeks post-last dose due to sperm life cycle * Known human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible; patients with compensated HIV, with adequate cluster of differentiation (CD)4+ T-cell counts, and not requiring antiretroviral medication will be allowed * Patients taking vitamin E supplements while on study * No concomitant anti-cancer chemotherapy or other systemic drugs; palliative radiation therapy will be allowed as long as the patient meets all other eligibility criteria * Refractory nausea and vomiting, chronic gastrointestinal diseases (e.g. inflammatory bowel disease), or significant bowel resection that would preclude adequate absorption * Patients with corrected QT (QTc) interval \> 450 msecs or other factors that increase the risk of QTc prolongation or arrhythmic events (e.g., heart failure, hypokalemia, family history of long QT interval syndrome) including heart failure that meets New York Heart Association (NYHA) class III and IV definitions are excluded * Every effort must be made to avoid the use of a concomitant medication that can prolong the QTc interval while receiving AZD6244; if the patient cannot discontinue medications that prolong the QTc interval while receiving AZD6244, close cardiac monitoring should be performed