I'm not a native English speaker, but my wife is. She doesn't know either. I think I've heard people use words "rag" and "rubbish", but I'm not sure how precise and widespread this is. If this matters, I'm interested in the word one would put on a package in a supermarket rather than use in plain everyday speech.
After some trial and error I found this article on Wikipedia: https://en.wikipedia.org/wiki/Floorcloth which suggests "floorcloth". But when I search Google for images, all I find is sorts of carpets, and nothing about cleaning.
Best Answer
Hardware stores in the U.S. sell packages of "new rags." Not surprisingly, they are often named "bag-o-rags" or "box-o-rags." Colloquially, "rag" is what the majority of native English speakers would use to describe a "cloth used to mop the floor".
I'd never use "floorcloth" and wouldn't have thought of it in 1,000 years. When I first read the compound word in your question, "rug" was the first thing that came to mind. A rapid search on the internet confirms this perspective. It's the wrong rabbit to chase.
To my knowledge, there is no formal word for what you're looking for. Colloquially, you will hear:
Mop the floor with this rag.
Mop the floor with that cloth.
Mop the floor with a towel.
Therefore, I'm not surprised to find that Proctor and Gamble invented a brand name to take the place of these generally-negative words: Swiffer. The name, like aspirin, kleenex, xerox, and a host of other brand names, has become so commonly used that it is losing its brand status and becoming the defacto way of describing something-with-which-you-mop-the-floor-that-you-don't-want-to-call-a-rag.
However, if you tried to use it in a commercial context, I'd expect P&G to sue you for trademark infringement.
In a commercial context...
Mop cloth
Mopping towel
would do, especially since the context already exists with paper towels.