Thomas Logan was the name of the groundskeeper at the Howlett Estate. He was the father of Dog ( possibly Victor/Sabretooth ), and may or may not have been the father of James ( Wolverine ). It was implied Thomas had a previous affair/relationship with James' mother.
After Thomas Logan killed James' father, James manisfested his powers for the first time and attacked and killed Thomas with his claws ( much like in the movie ). James and Rose ( who was brought to the estate to be his companion ) ran away, with Rose watching out for James, much like Victor did in the movie.
When Rose and James made their way to a mining camp to look for a place to live and work, the camp foreman asked who James was. Since James was still traumatized and wasn't speaking, Rose made up the story that James was her cousin. When asked for his name, Rose panicked and said "Logan" to conceal James' true identity. Henceforth, James became known as Logan to the public. Only Rose ( and Dog ) knew his real name was James and called him that in private.