Everolimus and Brentuximab Vedotin in Treating Patients With Relapsed or Refractory Hodgkin Lymphoma

Inclusion Criteria: * Biopsy-proven relapsed (response to last treatment \> 6 months duration), refractory (no response to last treatment or response duration \< 6 months) or residual Hodgkin lymphoma; NOTE: re-biopsy is necessary unless the patient has had a previous biopsy \< 180 days prior to registration on this protocol with no intervening therapy and tissue is available for translational research studies * Eligible to receive standard brentuximab vedotin for relapsed Hodgkin lymphoma * Measurable disease by CT or magnetic resonance imaging (MRI) or the CT portion of the PET/CT: must have at least one lesion that has a single diameter of \>= 2 cm * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1, or 2 * Absolute neutrophil count (ANC) \>= 1000/uL * Hemoglobin (Hgb) \>= 9 g/dl * Platelet (PLT) \>= 75,000/uL * Serum total bilirubin within normal range (or =\< 1.5 x upper limit of normal \[ULN\] if liver metastases are present; or total bilirubin =\< 3.0 x ULN with direct bilirubin within normal range in patients with well documented Gilbert syndrome) * Aspartate aminotransferase (AST) =\< 1.5 x ULN * Alkaline phosphatase =\< 1.5 x ULN * Serum creatinine =\< 1.5 x ULN * Negative serum pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only * Provide informed written consent * Willing to return to Mayo Clinic for follow-up * Willing to provide blood and tissue samples for correlative research purposes * Willingness to take everolimus orally and maintain a pill diary Exclusion Criteria: * Any of the following * Pregnant women * Nursing women * Women of childbearing potential who are unwilling to employ highly effective contraception while on study treatment and for 6 months after the final dose of treatment; NOTE: women of childbearing potential are defined as all women physiologically capable of becoming pregnant * Men of childbearing potential who are unwilling to employ highly effective contraception while on study treatment and for 6 months after the final dose of treatment and should not father a child during this time; NOTE: men of childbearing potential are defined as all males physiologically capable of conceiving offspring * Candidate for known standard therapy for the patient's disease that is potentially curative * Therapy with myelosuppressive chemotherapy or biologic therapy \< 21 days prior to registration unless the patient has recovered from the nadir of the previous treatment to a level that meets the inclusion eligibility criteria of this protocol * Patients who have received any continuous or intermittent small molecule therapeutics (excluding monoclonal antibodies) =\< 5 effective half-lives prior to registration or who have not recovered from side effects of such therapy * Received wide field radiotherapy =\< 28 days or limited field radiation for palliation =\< 14 days prior to registration or who have not recovered from side effects of such therapy * Receiving corticosteroids \> 20 mg of prednisone per day (or equivalent); Note: the dose should be noted on the medication record each cycle * Persistent toxicities \>= grade 2 from prior chemotherapy or biological therapy regardless of interval since last treatment * Patients who have any severe and/or uncontrolled medical conditions or other conditions that could affect their participation in the study such as: * Symptomatic congestive heart failure of New York Heart Association class III or IV * Unstable angina pectoris, symptomatic congestive heart failure, myocardial infarction within 6 months prior to registration, serious uncontrolled cardiac arrhythmia or any other clinically significant cardiac disease * Severely impaired lung function as defined as spirometry and diffusion capacity of carbon monoxide (DLCO) that is less than 50% of the normal predicted value and/or oxygen (O2) saturation that is 88% or less at rest on room air * Uncontrolled diabetes as defined by fasting serum glucose \> 1.5 x ULN (Note: optimal glycemic control should be achieved before starting everolimus) * Active (acute or chronic) or uncontrolled severe infections * Other concurrent chemotherapy, immunotherapy, radiotherapy, or any ancillary therapy considered investigational (utilized for a non-Food and Drug Administration \[FDA\]-approved indication and in the context of a research investigation) * Known positivity for human immunodeficiency virus (HIV); Note: baseline testing for HIV is not required * Active hepatitis B or C with uncontrolled disease * Note: a detailed assessment of hepatitis B/C medical history and risk factors must be done at screening for all patients; hepatitis B virus surface antigen (HBsAg) and hepatitis C virus (HCV) ribonucleic acid (RNA) polymerase chain reaction (PCR) testing are required at screening for all patients with a positive medical history based on risk factors and/or confirmation of prior hepatitis B virus (HBV) infection * Other active malignancy requiring treatment that would interfere with the assessments of response of the lymphoma to protocol treatment * Inability to swallow or impairment of gastrointestinal function or gastrointestinal disease that may significantly alter the absorption of the drugs (e.g., ulcerative disease, uncontrolled nausea, vomiting, diarrhea, malabsorption syndrome or small bowel resection) that would preclude use of oral medications * Major surgery =\< 14 days prior to registration or have not recovered from side effects of such therapy * Treated with any hematopoietic colony-stimulating growth factors (e.g., granulocyte-colony stimulating factor \[G-CSF\], granulocyte-macrophage colony-stimulating factor \[GM-CSF\]) =\< 2 weeks prior to study registration; NOTE: erythropoietin or darbepoetin therapy, if initiated at least 2 weeks prior to study registration, may be continued * Pre-existing neuropathy of \>= grade 2 * Patients receiving strong inhibitors of cytochrome P450 family 3, subfamily A, polypeptide 4 (CYP3A4) * Use of the following strong or moderate inhibitors are prohibited =\< 7 days prior to registration * Boceprevir (Victrelis \[TM\]) * Clarithromycin (Biaxin, Biaxin XL) * Conivaptan (Vaprisol) * Itraconazole (Sporanox) * Ketoconazole (Nizoral) * Nefazodone (Serzone) * Posaconazole (Noxafil) * Telithromycin (Ketek) * Voriconazole (Vfend) * Use of the following inducers are prohibited =\< 12 days prior to registration * Bosentan (Tracleer) * Carbamazepine (Carbatrol, Epitol, Equetro \[TM\], Tegretol, Tegretol-XR) * Modafinil (Provigil) * Phenobarbital (Luminal) * Phenytoin (Dilantin, Phenytek) * Rifabutin (Mycobutin) * Rifampin (Rifadin) * St. John's wort