This study will compare three treatment regimens containing metaiodobenzylguanidine (MIBG) and compare their effects on tumor response and associated side effects, to determine if one therapy is better than the other for people diagnosed with relapsed or persistent neuroblastoma.

Inclusion Criteria:

• Patients must be > 24 months and < 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 frontline therapy frontline therapy with > 3 residual lesions on end-induction MIBG scan.
• 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 adequate heart, kidney, liver and bone marrow function. Patients who have bone marrow disease must still have adequate bone marrow function to enter the study.
• Patients must have a dose of unpurged peripheral blood stem cells is 2.0 x 106 viable CD34+ cells/kg available.

Exclusion Criteria:

• They have had previous I-131 MIBG therapy
• They have other medical problems that could get much worse with this treatment.
• They are pregnant or breast feeding.
• They have a history of a venous or arterial thrombosis that was not associated to a central line.
• They have active infections such as hepatitis or fungal infections.
• They have active diarrhea.
• They have had an allogeneic stem cell transplant (received stem cell from someone else)
• They can't cooperate with the special precautions that are needed for this trial.