I've just been handed multiple docs that I need to compile and end up with the chapter number in front of the page number (e.g., 3-47). My problem is that the people who created the doc named their headings things other than the built-in "Heading 1". Is there any way to add other headings to the drop-down list on the Page Number Format window? I really don't want to have to reassign all these headers.

If the headings are associated with some special font and/or paragraph formatting, you could search for such special formatting, select/highlight all instances, then apply a Heading style to them.

The headings have names. I'm trying to find out if there's a way to add these heading names to the drop-down list so I can select them for adding chapter numbers to part of the page numbers.

What do you mean by "headings have names"?
Are these headings numbered in any way?

The headings are already assigned style names. I want different style names to show up in the drop down menu for formatting page numbers.

As far as I know, chapter numbers can only be inserted through the "Heading"-type styles associated with them. Sorry I could not be of any more help…
We might help more if you posted a sample document showing the kind of headings that have been used in your document(s).