The French game name jeu de dames was borrowed by Spanish as juego de damas. Of course, Spanish draughts players publishing a book mentioned this name in the title, like in these four ones:
N.N. “El ingenio à juego de marro, de punta, ò damas” (1547)
Pedro Ruiz Montero “Libro del juego de las damas vulgarmente nombrado el marro” (1591)
Lorenço Valls “Libro del juego de las damas, por otro nombre el marro de punta” (1597)
Garcia Canalejas “(…) las Damas, que vulgarmente llaman marro” (1650).
Each of these authors gives a second name for draughts. This should be an older name of the game.
Italian
The old Italian name of draughts was marella or a variant. The Sicilian Pietro Carrera in his work on chess, “Il gioco de gli scacchi” (1617) said about the name of draughts: “quel gioco, che i siciliani chiamano marella, e gli spagnuoli el juego de las damas” (the Sicilians call draughts marella, the Spaniards juego de las damas). On Sicily, the name marella was still used when players in Rome called draughts dama, see Michele Pasqualino in his dictionary “Vocabularo etimologico Siciliano” (1789): “Maredda di dudici (…), in Roma si dice a Dama” (maredda with [2x] twelve pieces (…), in Rome called dama).
There is also a game maredda played with 2×9 pieces: “Marredda di novi”, morris.
French
Where Spanish had the word marro and Italian marella or a variant, the French form was merelles. Two references from dictionaries: Sasbout 1579: “le jeu de merelles = damspel”; Hulsius 1631: “le ieu de merelles = Damspiel, Gioco di dama, Scruporum ludus”.
In the 14th c., the poet Jean Lefèvre wrote a long French poem entitled “La veille”. Just like the Italian Pasqualino he discerned merelles de neuf and merelles de douze (merelles with 2×9 and with 2×12 pieces). His text is an almost word-for-word translation from a Latin poem of his 13th c. compatriot Richard de Fournival (1201-c. 1260), author of popular romances of chivalry. “Young ladies and young men”, said Lefèvre, “are playing games called merelles where I can tell nothing new about. These games are only interesting when played with 2×9 of 2×12 pieces. The game with 2×9 pieces can be played with dice. The game with 2×12 pieces is played without dice, and the players takes by jumping over an enemy piece”.
If the medieval Italian and Frenchman could use the forms marredda di novi and merelles de neuf (marella/merelles with 2×9 pieces) and marredda di dudici and merelles de douze (marella/merelles with 2×12 pieces) it is obvious to seek the etymology of marella and merelles in a word meaning gaming piece. This word is Latin marrus, meaning stone, gaming piece.
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