This phase I/II trial studies the side effects and the best dose of brentuximab vedotin when given together with gemcitabine hydrochloride and to see how well they work in treating younger patients with relapsed or refractory Hodgkin lymphoma. Monoclonal antibodies, such as brentuximab vedotin, can block cancer growth in different ways. Some block the ability of cancer cells to grow and spread. Others find cancer cells and help kill them or carry cancer-killing substances to them. Drugs used in chemotherapy, such as gemcitabine hydrochloride, work in different ways to stop the growth of cancer cells, either by killing the cells or by stopping them from dividing. Giving brentuximab vedotin together with gemcitabine hydrochloride may kill more cancer cells.