Genetically Modified Peripheral Blood Stem Cell Transplant in Treating Patients With HIV-Associated Non-Hodgkin or Hodgkin Lymphoma

Inclusion Criteria: * HIV seropositive * Antiretroviral treatment for at least one month, defined as a multi-drug regimen (excluding azacitidine \[AZT\]) * HIV plasma viral load has decreased by 1.5 logs or viral load \< 5000 copies/ml * Non-Hodgkin or Hodgkin lymphoma without active central nervous system (CNS) involvement associated with poor prognosis with medical therapy alone or for which autologous peripheral blood stem cell (PBSC) transplant is indicated: * Hodgkin's lymphoma beyond first remission; first partial remission; induction failure with subsequent response to salvage therapy * Non-Hodgkin's Lymphoma beyond first remission: first partial remission; induction failure with subsequent response to salvage therapy * Chemotherapy responsive disease * Karnofsky performance score \>= 70% * Subjects must agree to use effective contraception from enrollment through completion of the study * Female subjects: if of child bearing potential, must have negative serum or urine pregnancy test within 7 days of treatment * Subjects must be on a prophylactic regimen for Pneumocystis carinii pneumonia, or agree to begin such treatment, if the cluster of differentiation (CD)4 counts are =\< 200 * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Serum creatinine \> 2 times upper limit of normal * Serum bilirubin greater than 3 times the upper limits of normal, unless determined to be a result of the primary hematologic malignancy or attributed to Gilbert's syndrome * Alanine aminotransferase (ALT) or aspartate aminotransferase (AST) greater than 3 times the upper limits of normal, unless determined to be a result of the primary hematologic malignancy or attributed to Gilbert's syndrome * Forced vital capacity (FVC), forced expiratory volume in 1 second (FEV1) or diffusion capacity of the lung of carbon monoxide (DLCO) parameters \< 60% predicted (corrected for hemoglobin) * Left ventricular ejection fraction (LVEF) \< 50% or coronary artery disease requiring treatment * Active infection requiring systemic antibiotic therapy with antibacterial, antifungal, or antiviral agents (excluding HIV) * Patients who are hepatitis C virus (HCV) antibody positive or hepatitis B virus (HBV) surface antigen positive must be free of clinical evidence of cirrhosis that would otherwise make them ineligible for HCT, as determined by the Principal Investigator (P.I.) in consultation with the Gastrointestinal Service; patients with HBV and ongoing evidence of viral replication may require therapy prior to receiving high-dose chemotherapy * Positive serology for Toxoplasma gondii AND requiring treatment or with evidence of active infection * Malignancy other than lymphoma, unless 1) in complete remission and more than 5 years from last treatment), or 2) cervical/anal squamous cell carcinoma in situ or 3) superficial basal cell and squamous cell cancers of the skin * History of HIV-associated encephalopathy; dementia of any kind; seizures in the past 12 months; any perceived inability to directly provide informed consent (Note: Consent may not be obtained by means of a legal guardian) * A medical history of noncompliance with highly active anti-retroviral therapy (HAART) or medical therapy