PRIMARY OBJECTIVES: I. To evaluate the safety and efficacy of a preparative regimen of Yttrium-90 (90^Y)- labeled anti-cluster of differentiation (CD)20 monoclonal antibody (MAb) (yttrium Y 90 ibritumomab tiuxetan) in combination with fludarabine (fludarabine phosphate) and melphalan followed by allogeneic hematopoietic stem cell transplant (APBSCT) for treatment of patients with B-cell low-grade non-Hodgkin lymphoma (LG NHL), intermediate-grade non-Hodgkin lymphoma (IG NHL) and mantle cell lymphoma (MCL). II. To evaluate the short- and long-term complications of this new preparative regimen, including rates of engraftment, acute and chronic graft-versus-host-disease (GVHD) and infectious complications. III. To estimate the disease response rate, disease relapse (progression) rate, and non-relapse mortality rate. IV. To perform exploratory studies that seek to measure/characterize the expression of costimulatory molecules and impact of these molecules on the natural killer (NK) and T cells of a subset of lymphoma patients pre- post- allogeneic stem cell transplant (ASCT) and the stem cell product from a portion of sibling donors.