Veliparib, Bendamustine Hydrochloride, and Rituximab in Treating Patients With Relapsed or Refractory Lymphoma, Multiple Myeloma, or Solid Tumors

Inclusion Criteria:

* Phase 1b: Patients must have a histologically confirmed solid malignancy, lymphoma or multiple myeloma for which standard curative or palliative measures do not exist, are no longer effective, or for which the patient is not eligible or refuses; phase 1b cohort expansion: patients must have a histologically confirmed cluster of differentiation (CD)-20 positive B-cell non-Hodgkin lymphoma for which standard curative or palliative measures do not exist, are no longer effective, or for which the patient is not eligible or refuses; phase 2a: patients must have histologically or cytologically confirmed marginal zone B-cell lymphoma, small lymphocytic lymphoma, lymphoplasmacytic lymphoma or mantle cell lymphoma, and must have at least one measureable site of disease
* For lymphoma and multiple myeloma patients: patients who have relapsed or are refractory to at least one prior chemotherapeutic regimen or biologic agent; patients must either be ineligible for or have refused curative options for treatment, including stem cell transplant, if applicable
* For solid tumor patients: relapsed or refractory to at least one prior chemotherapeutic regimen or biologic agent; patients must either be ineligible for or have refused curative options for treatment
* Patients must have had a rest period of at least 3 weeks since prior chemotherapy or radiation therapy, 6 weeks if the last regimen included carmustine (BCNU) or mitomycin C; there must be a rest period of at least 3 months if the last therapy was immunotherapy or radioimmunotherapy (unless the disease has progressed since treatment)
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%)
* Life expectancy of greater than 3 months
* Absolute neutrophil count (ANC) \>= 1,000/mcL
* Platelets \>= 100,000/mcL unsupported by transfusion within the prior 2 weeks
* Hemoglobin \>= 8.0 g/dL unsupported by transfusion within the prior 2 weeks
* Total bilirubin =\< 2 x upper normal institutional limits; in patients with Gilbert's disease or documented liver metastases, total bilirubin up to 3 x upper limits of normal (ULN) will be allowed
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 X institutional upper limit of normal
* Creatinine within normal institutional limits OR creatinine clearance \>= 60 mL/min for patients with creatinine levels above institutional normal
* Patients with prior stem cell transplant will be eligible as long as they have not relapsed or progressed within 100 days post-transplant and they meet the above inclusion criteria
* Women of child-bearing potential and men must agree to use adequate contraception (hormonal, barrier method of birth control, or abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately
* Ability to understand and the willingness to sign a written informed consent document
* Toxicities from prior therapies must have resolved to baseline, or be =\< grade 2 and stable for at least one month
* Patient must be able to swallow pills
* Patients with central nervous system (CNS) metastases must be stable after therapy for \> 3 months and off steroid treatment prior to study enrollment

Exclusion Criteria:

* Patients who have had chemotherapy or radiotherapy within 3 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered to baseline (or are not at stable grade =\< 2) from adverse events due to agents administered more than 3 weeks earlier; patients who have received immunotherapy or radioimmunotherapy within 3 months, unless disease has progressed since treatment; patients who have been administered ABT-888 as part of a single or limited dosing study, such as a phase 0 study, will not be excluded from participating in this study solely because of receiving prior ABT-888
* Patients may not be receiving any other investigational agents
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to ABT-888, bendamustine or mannitol; patients enrolling in the cohort expansion or phase 2 portions of the study who have been intolerant of repeated doses of rituximab in the past will be excluded (patients who have had infusion reactions to their initial dose of rituximab will not be excluded)
* Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
* Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated with ABT-888
* Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are eligible if their HIV is under adequate control with an antiretroviral regimen that has been stable for \>= 4 weeks, as long as the CD4 count is \> 300; appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated; patients on zidovudine or stavudine would not be eligible
* Patients with active seizure or a history of seizure are not eligible
* Patients with uncontrolled CNS metastasis are not eligible
* Patients with unrelated prior malignancies must have undergone potentially curative therapy for their prior malignancy, have no evidence of that disease for three years, or be deemed at low risk for recurrence of their prior malignancy by her/his treating physician; patients with dermal squamous cell carcinoma, basal cell carcinoma or melanoma in situ that has been completely excised will be eligible following excision