This pilot phase I trial studies the side effects and best dose of human immunodeficiency virus (HIV)-resistant gene modified stem cells in treating HIV-positive patients who are undergoing first-line treatment for Hodgkin or Non-Hodgkin Lymphoma. Stem cells are collected from the patient and HIV-resistance genes are placed into the stem cells. The stem cells are then re-infused into the patient. These genetically modified stem cells may help the body make cells that are resistant to HIV infection.