The stone’s name passed from Greek through Latin and French.
Greek marmaros entered Latin as marmor. Anglo-French marbre then reached English through Middle English. The modern word looks much closer to the French form than to the earlier pair.
Middle English is an omitted intermediate stage. The question concerns the stone name, not the later game pieces.
Source
Merriam-Webster: marbleThe noun's Word History supplies Greek, Latin, Anglo-French and Middle English stages.