What number determines the name of an atom?
The number of protons in the nucleus determines the name of the atom.
The properties of an atom are fundamentally defined by the number of protons in the nucleus. Once this was recognized, elements were assigned names according to the unique, integer values of protons they possessed.
As straightforward as this is, we should recognize that this is a connection between empirical facts and subjective categories, and that there is no fundamental link between numbers of protons and names of the atoms, whereas we might expect there to be a link between numbers of protons and the total charge, or numbers of protons and baseline mass. Ultimately the thing that determines the name of the atom is human choice; we just happen to have linked it to the number of protons. We might just as easily have come up with a unique name for every single isotope, and then said that the atomic mass and charge determines the name.