This is called the Footnote Separator. There is also a Footnote Continuation
Separator, which, by default, does go all the way across the page; it
appears only when a footnote is continued to the next page.

The way to access either of these for editing is to switch to Normal view,
then View | Footnotes. There is a dropdown list on the menu bar from which
you can select Footnote Separator. Try pressing the Reset button to see if
this brings the usual separator back. If that doesn't work, I don't know
what recourse you have except to construct it by hand using underlines or em
dashes. If you want to delete it, that's easily enough done: select it and
press Delete. But whatever choice you make is global: it will apply to *all*
footnotes in that document.

--
Suzanne S. Barnhill
Microsoft MVP (Word)