This is a subject I want to explore in some depth. The Obama FightTheSmears web site published a diagram worthy of the best conspiracy theorist about just where the stories come from, tracking some back to the 2004 campaign against John Kerry.
Others blame Andy Martin for a a chain email starting the story that Barack Obama was a Muslim. Martin was labeled by the New York Times as “The Man Behind the Whispers About Obama.”
The smears persist thanks to several interest groups including pro-Israel, anti-Muslim, white racist, anti-abortion, PUMA (pro-Hillary), anti-federalism, pro-gun and right-wingers in general. How much cooperation there is; beyond cutting and pasting each others’ material, I don’t know.