Randomized Phase II Study of AZD6244 (Mitogen-activated Protein Kinase Inhibitor) MEK-Inhibitor With Erlotinib in KRAS Wild Type Advanced Non-Small Cell Lung Cancer (NSCLC) and a Randomized Phase II Study of AZD6244 With Erlotinib in Mutant KRAS Adva...

* INCLUSION CRITERIA: * Patients must have histologically or cytologically confirmed advanced (Stage IV) Non Small Cell Lung Cancer that has been verified by the enrolling institution s pathology department. Mixed histologies, with small cell lung cancer components are not eligible. * Patients must have measurable disease by RECIST criteria, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as greater than or equal to 20 mm with conventional techniques or as greater than or equal to 10 mm with spiral CT (computed tomography) scan. * Patients must have had at least one prior platinum-containing chemotherapy regimen, or have refused cytotoxic chemotherapy. Patients should have no more than two prior chemotherapy regimens. Chemotherapy regimens have no limit on the maximum or minimum number of cycles. Patients enrolled on the trial should have completed their last chemotherapy regimen at least 4 weeks ago. Patients should have completed radiation therapy at least 4 weeks prior to enrollment. Adjuvant and neoadjuvant chemotherapy does not count as prior chemotherapy for advanced disease if more than a year before enrollment. * Age greater than or equal to 18 years, because no dosing or adverse event data are currently available on the use of AZD6244 (ARRY-142886) as monotherapy or in combination with erlotinib in patients less than 18 years of age. Children are excluded from this study but will be eligible for future pediatric phase 2 combination trials. * Life expectancy of greater than 3 months. * ECOG (Eastern Cooperative Oncology Group) performance status less than or equal to 2 (Karnofsky greater than or equal to 60%). * Patients must have normal organ and marrow function as defined below: * leukocytes greater than or equal to 3,000/mcL * absolute neutrophil count greater than or equal to 1,500/mcL * platelets greater than or equal to 100,000/mcL * total bilirubin within normal institutional limits * AST (aspartate aminotransferase) (SGOT (serum glutamic oxaloacetic transaminase)) /ALT (alanine aminotransferase) ((SGPT (serum glutamic pyruvic transaminase)) less than or equal to 2.5 X institutional upper limit of normal * creatinine within normal institutional limits OR * creatinine clearance greater than or equal to 60 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * Patients must have adequate archival material from a previous biopsy to determine KRAS status at the time of enrollment, or undergo a biopsy of fresh tissue of the primary cancer lesion or a metastatic site in order to make this determination. * The effects of AZD6244 and erlotinib on the developing human fetus at the recommended therapeutic dose are unknown. However preclinical data showing adverse effects on fetal survival and development with AZD6244 have been reported. For this reason since there is a potential risk of teratogenicity, 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, and for four weeks after dosing with AZD6244 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 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. * Ability to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA: * Patients who have had chemotherapy or radiotherapy within 4 weeks prior to entering the study or those who have not recovered to less than or equal to grade 1 toxicities from adverse events due to agents administered more than 4 weeks earlier. * Patients may not be receiving any other investigational agents. * In general, 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. However, patients who have completed primary treatment of their brain metastasis by surgery or radiotherapy and have been off steroids (with the exception of maintenance replacement steroids) and anti-seizure medications for greater than one month without progression of neurological symptoms are eligible for enrollment in this trial. Stability of treated brain metastases should be confirmed by repeat imaging studies (CT brain or MRI (magnetic resonance imaging) brain) performed at least one month after completion of treatment. * Previous MEK (methyl ethyl ketone) inhibitor or prior treatment with EGFR TKI (tyrosine kinase inhibitor). * Major surgery or significant traumatic injury occurring within 21 days prior to treatment. * Abnormalities of the cornea based on history (e.g., dry eye syndrome, Sjogrens syndrome), congenital abnormality (e.g., Fuchs dystrophy), abnormal slit-lamp examination using a vital dye (e.g., fluorescein, Bengal-Rose), and/or an abnormal corneal sensitivity test (Schirmer test or similar tear production test). * Gastrointestinal tract disease resulting in an inability to take oral medication or a requirement for IV (intravenous) alimentation, prior surgical procedures affecting absorption, or active peptic ulcer disease. * Patients with uncontrolled hypertension already on optimal medication, patients with class II or greater heart failure, any current or prior history of cardiomyopathy. Patient with baseline LVEF (left ventricular ejection fraction) less than 50%, atrial fibrillation, recent myocardial infarction, or unstable ischemic heart disease. * Patients with QTc (corrected QT interval) interval greater than 450 msecs or other factors that increase the risk of QT (Q wave, T wave) 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. This does not include QTc prolongation secondary to a paced rhythm. If a patient has a paced rhythm, QTc levels less than or equal to 500 (Grade 1) are allowed and patients with QTc \> 500 are excluded. * Required use of a concomitant medication that can prolong the QT interval. A comprehensive list of agents with the potential to cause QTc prolongation can be found at http://www.arizonacert.org/medical-pros/drug-lists/drug-lists.htm. * Refractory nausea and vomiting, chronic gastrointestinal diseases (e.g. inflammatory bowel disease), or significant bowel resection that would preclude adequate absorption. * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, Hepatitis B or C, 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 AZD6244 is a MEK- Inhibitor agent with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with AZD6244, breastfeeding should be discontinued if the mother is treated with AZD6244. These potential risks may also apply to other agents used in this study. * HIV (human immunodeficiency virus) -positive patients on combination antiretroviral therapy are ineligible because of the potential for pharmacokinetic interactions with AZD6244. In addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy. Appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated. INCLUSION OF WOMEN AND MINORITIES: Both men and women and members of all races and ethnic groups are eligible for this trial.