Is it permissible to name another son "Sulayman" after the death of a previous son who bore the same name, given that the deceased son was one month old?

There is no harm in naming several children with the same name, even if they are all alive, let alone if the first one passed away. This is indicated by the absence of any evidence prohibiting it, and by the fact that Umar ibn al-Khattab named three of his sons Abd al-Rahman.