Twice, while my wife was working editing a Word file, paragraph marks appeared, unsolicited, in the document.

I was able to remove them by using
File> Options > Display
where I found the box for formatting symbols was checked, so I unchecked it.

She must have invoked this feature by mistyping something, but I have no idea what it was. Can anyone give me a clue?

First, if she cliced on the pillcrow (paragraph mark) on the formatting toolbar, that will display them. This is a toggle. Clicking it again will hide them.

There is a keyboard shortcut for toggling display using Ctrl+Shift+8 or Ctrl+*. These are not the keys in the numeric keypad but the regular typing keyboard. This shortcut works in Word 97-Word 2013 (and probably earlier and later versions as well).

For more, see Showing non-printing formatting marks in Microsoft Word.