What is his purpose and why does Steinbeck put Noah in The Grapes of Wrath?
Functionally and superficially, Noah has several roles. We can also look at his character thematically.
Functionally, Noah is the eldest son and so should hold a certain position of authority in the family. He does not fulfill the role assigned to this position however as he is tempermentally incapable of doing so. He is just not a leader. In this way, Tom becomes the leader of his generation, having overcome this formal obstacle.
(Formal obstacles and arbitrary hurdles often seem to get between Tom and his goals. Overcoming them is part of his development and part of what makes him the protagonist of the novel.)
Superficially, Noah's presence brings the family number up to twelve (Ma, Pa, Uncle John, Noah, Tom, Rose of Sharon, Connie, Al, Winfield, Ruthie, Granpa, Granma). This has a Biblical resonance when considered alongside the presence of the preacher, Casy.
Thematically, Noah is an example of how the family breaks up under stress as well as under more overtly physical demands.
When [Tom] tells her about Noah, Ma feels the “family’s fallin’ apart.”
Death and despair reduce the family to eight by the time the Joads leave their first camp in California. Noah represents some of this despair, saying that he is "sad" and has to stay by the river.