Hydroxychloroquine With or Without Erlotinib in Advanced Non-small Cell Lung Cancer (NSCLC)

* The goal of this study is to find the highest dose of HCQ that can be given safely with erlotinib. Therefore, not all participants will receive the same dose of HCQ. Small groups of participants will be enrolled in steps in this trial. The first group will be given a certain dose of HCQ. If they have few or manageable side effects, the next small group of participants enrolled will receive a higher dose. This increase in doses will continue until the research doctors find the highest dose of HCQ that can be given without causing severe or unmanageable side effects. * Both HCQ and erlotinib are pills that are taken orally. Treatment will be divided into time periods called cycles. Each treatment cycle is 28 days. The exception to this 28 day cycle is when participants start taking the pills for the first time. Erlotinib is started first for 7 days and then HCQ is added. When the HCQ begins, the first cycle of 28 days begins. * There are several tests and procedures that will be performed at specific time periods during protocol treatment. These include: blood work, performance status assessment, questions about medical history and medications, tumor assessment with CT or MRI and, eye exams. * Participants may continue to receive study treatment as long as they do not experience unacceptable side effects or disease progression.