The one-armed Castle Black blacksmith from the novels who was omitted from Game of Thrones is Donal Noye.
Noye is a smith, armorer, and steward of the Night's Watch in A Song of Ice and Fire. Before joining the Watch, he served House Baratheon at Storm's End and made Robert Baratheon's warhammer. He lost his left arm after an axe wound became infected during the siege of Storm's End.
At Castle Black, Noye helps train and equip the Night's Watch. He also recognizes Jon Snow's unusual skill with a sword and later fights the giant Mag the Mighty during the wildling assault on the Wall, sacrificing himself to protect the Watch. His absence from the television series is easy to confuse with the show's inclusion of other Castle Black officers, such as Alliser Thorne and Othell Yarwyck.
Noye is not the same character as Gendry, the apprentice smith shown in the series. Both are associated with blacksmithing, but Gendry is a young former apprentice from King's Landing, while Noye is an experienced Night's Watch veteran from the novels.