The plot thickens.
I figured maybe I could get the menu item to disappear by resizing my desktop. I did this using right-click on the desktop background, Properties, Settings, adjust screen resolution slider, Apply.
Since this was meant to clear the menu item but not resize the display permanently, I let the timer expire, and the desktop was resized back to where it was originally. However, now the icons were arranged as for the smaller screen. It seemed to work, but here's where the weird stuff happened.
I right-clicked the desktop background, intending to rearrange my icons. When I hovered over "Arrange Icons By" the menu I expected to pop up appeared grey as if inaccessible. But it was accessible, and I selected Name. Now the Name menu item is "permanently" displayed over my desktop like the menu item in my original post was.