Merels
English, from Old French · Britain, and the medieval record
Merels, merrills, morris
Before it settled into Nine Men's Morris, the game went by merels and merrills in English, and it still does in older books. The word is generally traced to the Old French merel, meaning a counter or token — the piece you play with, rather than anything to do with the game's shape.
That etymology is also the most likely explanation of the odd word in the modern name. The morris in Nine Men's Morris is probably the same merel worn down by centuries of English mouths, and not a reference to morris dancing, however tidy that story sounds. This is disputed rather than settled, and anyone who tells you it is certain in either direction is overselling it.
Boards cut into the furniture
The archaeological record for this game is unusually good, because people carved the board wherever they were sitting. Mill boards turn up scratched into cathedral cloister benches, castle window seats, and the stone of buildings whose builders presumably had time to fill. It is one of the few games where the evidence of ordinary people playing it survives in the fabric of the buildings themselves.
It was also cut into turf. A board scraped into the grass of a village green is the version Shakespeare had in mind in A Midsummer Night's Dream, where a season of bad weather leaves the board unplayable: "The nine men's morris is filled up with mud." The line is a complaint about the summer, and it only works because the audience knew exactly what the board was.
Why the old name is worth keeping
Merels is useful because it names the family rather than one member of it. Three, six, nine and twelve-piece boards are all merels; only one of them is Nine Men's Morris. If you are reading anything written before the twentieth century, or anything about the medieval game, merels is the word you will meet.
The rules under the old name are the rules you already know. Nine pieces each, place them, then slide them along the lines; three in a row takes an enemy piece; you lose at two pieces or when you cannot move.
The full rules are on how to play, the tactics on strategy, and the other boards on variants.