Institutions, companies, research centers and universities are paying special attention to green hydrogen, a key energy carrier for the decarbonization of various sectors because it is produced from renewable sources without producing emissions. Currently, however, the cost of producing it is high and does not make it competitive with that produced from fossil fuels. One way to cut costs could be to exploit the oxygen obtained as a by-product of water electrolysis, selling it instead of releasing it into the atmosphere as is usually done. In some studies, this idea has already been analyzed, but very often without considering the costs due to the additional components for treating the oxygen, which must be properly compressed and stored, or without directly showing how the LCOH (levelized cost of hydrogen) varies according to this choice, especially in an Italian context, limiting the study to an evaluation of the NPV. The paper analyzes plant configurations for the production of green hydrogen installed in southern Italy, in Sicily, in which a photovoltaic system powers an alkaline electrolyzer and two compressors: one for hydrogen and one for oxygen. The system is completed by the storage systems for the two gases. Levelized cost values for green hydrogen are calculated by considering the sale of oxygen either at prices found in the literature in similar studies, or at prices in the European Union for oxygen produced by air distillation, taking into account the operating and investment costs for the compressor and oxygen storage, and comparing the values obtained with the costs of green hydrogen without oxygen valorization.