Bendamustine Hydrochloride, Clofarabine, and Etoposide in Treating Younger Patients With Relapsed or Refractory Hematologic Malignancies

INCLUSION CRITERIA * Participants with Hodgkin or Non-Hodgkin lymphoma must meet one of the following criteria: (a) Relapsing disease in 2nd or greater relapse and measurable disease, or (b) Refractory disease failing to achieve complete remission (CR) with \> 2 induction or re-induction attempts. * Participant with acute leukemia must meet one of the following criteria: (a) Relapsing acute lymphoblastic leukemia (ALL), acute myeloid leukemia (AML), or acute biphenotypic leukemia in 2nd or greater relapse; or (b) Refractory ALL, AML, or acute biphenotypic leukemia failing to achieve CR with ≥ 2 induction or re-induction attempts. * Participant with leukemia has M2 or M3 marrow at the time of enrollment. Participant with M2 marrow must have definite cytogenetic, molecular, or immunophenotypic evidence of recurrent/refractory disease. * Age is ≤ 21 years (participant has not yet reached 22nd birthday). * Karnofsky or Lansky performance score is ≥ 60%. The Lansky performance score should be used for participants \< 16 years and the Karnofsky performance score for participants ≥ 16 years. * There are no known contra-indications to any of the planned agents used in this protocol. Etoposide may be substituted by etoposide phosphate (etopophos) if the patient has a history of hypersensitivity reaction to etoposide * Adequate renal function defined as glomerular filtration rate \> 60 cc/min/1.73m2, or normal serum creatinine based on age. * Adequate hepatic function: (a) Direct bilirubin ≤ upper limit of normal (ULN) for age, or if total bilirubin is \> ULN, direct bilirubin is ≤ 1.4 mg/dl, and (b) AST and ALT ≤ 5 x ULN for age. * Adequate cardiac function defined as shortening fraction of ≥ 27% or ejection fraction ≥ 45%. * Lymphoma participants without bone marrow involvement must have: (a) Absolute neutrophil count (ANC) ≥ 1,000/µL, and (b) Platelet count \> 50,000/mm\^3 (without transfusion support). \[Note: these criteria are waived for participants with leukemia or lymphoma participants with bone marrow involvement.\] * Participant must have recovered from the acute side effects of all prior anti-cancer therapy, and : * At least 2 weeks have elapsed since prior systemic cytotoxic chemotherapy (except intrathecal chemotherapy, and/or low dose maintenance therapy such as vincristine, mercaptopurine, methotrexate or glucocorticoids), and * At least 4 weeks have elapsed since treatment with an investigational agent or antibody-based therapy, if applicable, and * If the participant received a prior allogeneic hematopoietic stem cell transplantation (HSCT), at least 3 months have elapsed and there is no evidence of active graft-versus-host disease (GVHD), participant has discontinued immunosuppression, and there is no history of veno-occlusive disease. EXCLUSION CRITERIA * Active, uncontrolled infection or severe concurrent medical disease, including but not limited to congestive heart failure, cardiac arrhythmias, or psychiatric illness. * Isolated extramedullary disease (leukemia). * Primary CNS lymphoma. * Pregnant or lactating (female participant of childbearing potential must have negative serum or urine pregnancy test required within 7 days prior to start of treatment). * Known HIV or active hepatitis B or C infection. * Known hypersensitivity to bendamustine or mannitol.