Greenholm was a town or village of the Hobbits on the Far Downs, which originally lay on the western borders of the Shire. It was the home of Fastred, who married Sam Gamgee's daughter Elanor, and with her founded the Fairbairn Family. When King Elessar extended the Shire's borders westward to the Tower Hills, Fastred and Elanor removed from Greenholm into the new Westmarch of the Shire.
The name Greenholm is open to various interpretations, because the old word 'holm' has a number of possible meanings. It might possibly refer to a river-island, or even to the holly tree, but a much more likely explanation for the name, especially as it lay on the Far Downs, would be simply 'green hill'.