If an element has more than one ionic charge, how is that piece of information represented in a chemical name?

This only occurs with the ions formed by the transition metals. For example, Copper can form ions with a ${1}^{+}$ or a 2^+" charge. You would indicate the charge on the ion with a Roman numeral.
Consider $\text{CuCl}$. Since the chloride ion always has a 1^(-)" charge, and there are no subscripts, we can know that the copper ion in this compound has a charge of 1^+" in order for the compound to be neutral. We would name this compound copper(I) chloride, or in words only, copper one chloride. $\text{CuCl}$ can also be named cuprous chloride, though it is not used very often today.
Now consider $\text{CuCl"_2}$. There are two chloride ions and one copper ion. The total chloride charge is 2xx1^(-)=2^(-)". The copper ion does not have a subscript and we know that ionic compounds are neutral. So in order for $\text{CuCl"_2}$ to be neutral, the charge on the single copper ion is 2^+". We would name this ionic compound copper(II) chloride, or in words only, copper two chloride. This compound can also be named cupric chloride.