Busulfan, Melphalan, and Stem Cell Transplant After Chemotherapy in Treating Patients With Newly Diagnosed High-Risk Neuroblastoma

Inclusion Criteria: * Patients must have a diagnosis of neuroblastoma (International Classification of Diseases for Oncology \[ICD-O\] morphology 9500/3) or ganglioneuroblastoma (nodular or intermixed) verified by histology or demonstration of clumps of tumor cells in bone marrow with elevated urinary catecholamine metabolites; patients with the following disease stages at diagnosis are eligible, if they meet the other specified criteria * Patients with newly diagnosed neuroblastoma with International Neuroblastoma Staging System (INSS) stage 4 are eligible with the following: * V-myc avian myelocytomatosis viral oncogene neuroblastoma derived homolog (MYCN) amplification (\> 4-fold increase in MYCN signals as compared to reference signals), regardless of age or additional biologic features or * Age \> 18 months (\> 547 days) regardless of biologic features or * Age 12-18 months (365-547 days) with any of the following 3 unfavorable biologic features (MYCN amplification, unfavorable pathology and/or deoxyribonucleic acid \[DNA\] index = 1) or any biologic feature that is indeterminate/unsatisfactory/unknown * Patients with newly diagnosed neuroblastoma with INSS stage 3 are eligible with the following: * MYCN amplification (\> 4-fold increase in MYCN signals as compared to reference signals), regardless of age or additional biologic features or * Age \> 18 months (\> 547 days) with unfavorable pathology, regardless of MYCN status * Patients with newly diagnosed neuroblastoma with INSS stage 2A/2B with MYCN amplification (\> 4-fold increase in MYCN signals as compared to reference signals), regardless of age or additional biologic features * Patients with newly diagnosed neuroblastoma with INSS stage 4S with MYCN amplification (\> 4-fold increase in MYCN expression signals as compared to reference signals), regardless of additional biologic features * Patients \>= 365 days initially diagnosed with neuroblastoma INSS stage 1, 2, 4S who progressed to a stage 4 without interval chemotherapy; these patients must have been enrolled on ANBL00B1; study enrollment on ANBL12P1 must occur within 4 weeks of progression to stage 4 for INSS stage 1, 2, 4S * Patients must not have had prior systemic therapy except for localized emergency radiation to sites of life-threatening or function-threatening disease and/or no more than 1 cycle of chemotherapy per a low or intermediate risk neuroblastoma regimen (as per P9641, A3961, ANBL0531, or similar) prior to determination of MYCN amplification status and histology * Creatinine clearance or radioisotope glomerular filtration rate (GFR) \>= 70 mL/min/1.73 m\^2 or a serum creatinine based on age/gender as follows: * Age 1 month to \< 6 months: 0.4 mg/dL * Age 6 months to \< 1 year: 0.5 mg/dL * Age 1 to \< 2 years: 0.6 mg/dL * Age 2 to \< 6 years: 0.8 mg/dL * Age 6 to \< 10 years: 1 mg/dL * Age 10 to \< 13 years: 1.2 mg/dL * Age 13 to \< 16 years: 1.5 mg/dL (males), 1.4 mg/dL (females) * Age \>= 16 years: 1.7 mg/dL (males), 1.4 mg/dL (females) * Total bilirubin =\< 1.5 x upper limit of normal (ULN) for age, and * Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) or serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) \< 10 x ULN for age * Shortening fraction of \>= 27% by echocardiogram, or * Ejection fraction of \>= 50% by radionuclide evaluation * No known contraindication to peripheral blood stem cell (PBSC) collection; examples of contraindications might be a weight or size less than that determined to be feasible at the collecting institution, or a physical condition that would limit the ability of the child to undergo apheresis catheter placement (if necessary) and/or the apheresis procedure * All patients and/or their parents or legal guardians must sign a written informed consent * All institutional, Food and Drug Administration (FDA), and National Cancer Institute (NCI) requirements for human studies must be met Exclusion Criteria: * Patients that are 12-18 months of age with INSS stage 4 and all 3 favorable biologic features (ie, nonamplified MYCN, favorable pathology, and DNA index \> 1) are not eligible * Female patients who are pregnant are ineligible * Lactating females are not eligible unless they have agreed not to breastfeed their infants * Female patients of childbearing potential are not eligible unless a negative pregnancy test result has been obtained * Sexually active patients of reproductive potential are not eligible unless they have agreed to use an effective contraceptive method for the duration of their study participation