Ian James is the infant son of Sophie and Mark. The two are not married or even together, but they are friendly and polite to each other when it comes to raising Ian James. He is known mostly just as "baby Ian", and was named after his grandfather Ian, although Mark's first choice for a name was James Ian.
It seems as though Ian James was the reason for Sophie stopping her clubbing and wild behavior, as she had always wanted a child.
Originally, Sophie did not know whether her pregnancy was the result of sex with Mark, Jeremy, or Jeff. Eventually she took a paternity test before the baby was born to discover it was Mark's. However, Sophie remains friends with Jeff, who is possibly romantically interested in Sophie, and ends up spending more time with Ian James than Mark does.
Ian James' name is first discussed in the episode Jeremy at JLB when Sophie sees Mark after work. She tells Jeff that she found out the baby is a boy before she tells Mark, who only finds out through Jeff. They begin to think about names and Sophie suggests the name Tarquin Oliver Nimrod, to which Mark laughs and suggests "a name like Bruce". They argue over whether or not Bruce is a good name. Mark eventually says he doesn't even like the name Bruce, but he wants a normal name "like Bruce".
Ian James isn't officially named until the episode Nether Zone, when Mark misses his Christening due to being locked in a stranger's apartment lobby. He is against the name Ian James originally, but because he misses the Christening, his vote is not counted, and instead of his first choice, James Ian, the baby is named Ian James after his grandfather Ian, whom Mark both dislikes and is afraid of.