N2007-03: Vorinostat and 131-I MIBG in Treating Patients With Resistant or Relapsed Neuroblastoma

Inclusion Criteria: * Patients must be at least 24 months and no older than 30 years of age when registered on study. * Patients must have relapsed neuroblastoma, refractory neuroblastoma that had less than a partial response to standard treatment or persistent neuroblastoma that had at least a partial response to standard treatment. * Patients who have at least a partial response to standard treatment who still have neuroblastoma that can be seen on CT/MRI or MIBG scans must have a surgical biopsy done of the tumor to confirm that it is neuroblastoma. Patients with relapsed or refractory neuroblastoma do not need to have a biopsy done to enter on study. * Patients must have evidence of MIBG uptake into tumor at one site within 4 weeks prior to entry on study and subsequent to any intervening therapy. * Patients must have a stem cell product available that meets study criteria. If they don't already have stem cells frozen away then they must be able to have a stem cell collection done to collect the necessary amount of stem cells for study entry and these stem cells must meet study criteria. * Patients must have adequate heart, kidney, liver and bone marrow function. Patients who have bone marrow disease must meet the bone marrow function criteria to enter the study. Exclusion Criteria: * They have had treatment with 131I-MIBG before. * They have had prior treatment with vorinostat or other HDAC inhibitor. * They have had a stem cell transplant using another person as the stem cell donor. (You can still be in the study if a previous transplant used your own stem cells) * They have other medical problems that could get much worse if they had this treatment. * They are on dialysis for bad kidney function. * They have a history of unexplained blood clot, pulmonary embolus, thrombotic stroke, or arterial clot. * They are pregnant or breast feeding. * They have active infections such as hepatitis or fungal infections. * They had total body radiation or radiation to the entire belly or a large amount of radiation to the liver or kidney (some radiation to the liver or kidneys is ok). * They can't cooperate with the special precautions that are needed during MIBG treatment.