Ian James is the infant son of Sophie and Mark, born at St. Majella's Hospital in the first episode of series 7.
He is known mostly just as "baby Ian", and is ultimately named after his grandfather Ian. He makes his first appearance in the episode St Hospitals, when he is born. He isn't named, however, until the episode Nether Zone when Mark misses his Christening.
Paternity
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.
Naming
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".