This is a Phase I clinical trial evaluating the combination of hydroxyurea and magnesium pidolate for patients with sickle cell disease with either hemoglobin SS disease or hemoglobin S beta thalassemia. Hydroxyurea and magnesium pidolate will be tested in pediatric and adolescent patients with sickle cell disease who already have been treated with hydroxyurea for a minimum of six months. Magnesium pidolate will be given in combination with hydroxyurea for six months. In successive small groups of patients, the dose of magnesium will be increased in order to eventually determine the maximum tolerated dose (MTD) and dose limiting toxicity (DLT) for magnesium when given in combination with hydroxyurea. The maximum tolerated dose is the highest drug dose that can be given safely to participants. The dose limiting toxicity is determined when drug side effects prevent an increase in dose.