I saved an XLS file as an XLSX file without performing a conversion. After
making changes to the workbook, the chart names will not save, they just
revery back to the default "Chart1", "Chart2", etc. names.

I followed these steps to change the names to no avail:
When you create charts, Microsoft Office Excel assigns a default name to
each chart by using the following naming convention: Chart1, Chart2, and so
on. However, you can change the name of each chart to make it more meaningful
to you.

Click the chart that you want to rename.
This displays the Chart Tools, adding the Design, Layout, and Format tabs.

On the Layout tab, in the Properties group, click the Chart Name text box.
Tip You may have to click the Properties icon in the Properties group to
expand the group.

Type a new name.
Press ENTER.

Any idea on how to save the names properly?

Thanks,
Brett