See Politifact's later explanation on this from September 25, 2015: The Obama Muslim Myth: The Clinton connection.
In the fall of 2007, it was clear that Clinton was losing ground in Iowa. The presumptive front runner was vulnerable and people in the campaign knew it. On Dec. 5, 2007, the online magazine Politico posted the text of an email that had been forwarded by Judy Rose, the volunteer chair of the Clinton campaign in Jones County Iowa on Nov. 21, 2007. The email was a quintessential smear that offered a distorted biography of Obama’s early years. Rose offered no commentary on it. She simply passed it along.
"Obama takes great care to conceal the fact that he is a Muslim," the email said, and it ended with, "The Muslims have said they plan on destroying the U.S. from the inside out, what better way to start than at the highest level - through the President of the United States , one of their own!!!!"
Rose, sent this to eight of her fellow Democrats. One of them was Clinton campaign staffer Ryan Callanan who replied to the email on the same day in November.
"I’ve gotten this forward before," Callanan wrote. "It’s racist and ignorant. I can’t believe that people believe this stuff."
The public airing of the email brought a quick reaction from the Clinton campaign. On Dec. 6, 2007, the Associated Press reported that Rose had resigned as chair of her county committee.
"There is no place in our campaign, or any campaign, for this kind of politics," Clinton campaign manager Patti Solis Doyle said. "A volunteer county coordinator made the mistake of forwarding an outrageous and offensive chain e-mail. This was wholly unauthorized and we were totally unaware of it."
A few days later, a second volunteer in a different county stepped down when it was learned that she had forwarded a similar email in October.
These are the only two instances where Clinton campaign volunteers, much less staff, are known to have spread the false story about Obama’s religion.
Excerpt:
My inquiring mind wants to know. Has anyone seen Obama’s birth certificate? Just to ease my curious mind, I would like to see it.
Should his birth certificate be in the public domain? Or not? I do not have a clue.
Excerpt:
A chain e-mail that originates with a letter from American missionaries working in Kenya warns about Sen. Barack Obama's ties to Kenya and its opposition party, encouraging readers "not to be taken in by those that are promoting him."
Among the many allegations is one about Obama's name: "By the way. His true name is Barak Hussein Muhammed Obama. Won't that sound sweet to our enemies as they swear him in on the Koran! God bless you."
[snip]
Let's be clear: The senator from Illinois who is running for president of the United States is named Barack Hussein Obama Jr. His campaign has insisted that is his full name (no Muhammad). We've checked this before and found it to be blatantly wrong. But since the rumor persists, we decided to dig deeper.
We have now searched every public record we can access and find nothing to support the notion that Obama has a second middle name of "Muhammed" or "Mohammed." Only in the ether of chain e-mails does this allegation fly.
We have a copy of Obama's marriage certificate from the Cook County Bureau of Vital Statistics in Illinois. Barack H. Obama married Michelle L. Robinson on Oct. 3, 1992, in Chicago. (And, yes, the Rev. Jeremiah A. Wright Jr. officiated.)
We looked at Obama's driver's license record in the state of Illinois, and the name reads: Barack H Obama. (Senator, just a reminder that your license is up for renewal this year).
We searched property records for Obama and found listings under the names Barack Hussein Obama and Barack H Obama.
We also found Obama's registration and disciplinary record with the Attorney Registration & Disciplinary Commission of the Supreme Court of Illinois. ( You can see it here. ) Barack Hussein Obama was admitted as a lawyer by the Illinois Supreme Court on Dec. 17, 1991. (By the way, Obama has no public record of discipline.)
We tried to obtain a copy of Obama's birth certificate, but his campaign would not release it and the state of Hawaii does not make such records public.
This was the spark that lit the fuse.
Excerpt:
Having done some Obama-rumor debunking that got praise from Daily Kos (a sign of the apocalypse, no doubt), perhaps the Obama campaign could return the favor and help debunk a bunch of others with a simple step: Could they release a copy of his birth certificate?
Reporters have asked for it and been denied, and the state of Hawaii does not make such records public.
Excerpt:
Jim Geraghty takes a look at longstanding blog buzz over Barack Obama’s birth certificate, which the campaign refused to release to the St. Petersburg Times in April:
Has anyone seen it? Why shouldn’t the record be in the public domain for presidential candidates?
We tried to obtain a copy of Obama’s birth certificate, but his campaign would not release it and the state of Hawaii does not make such records public.
Geraghty walks through various rumors now circulating in the wake of the Obama campaign’s birth certificate blackout...
In the information vaccuum create by Politifact and the Obama campaign, the internet swooped in to fill the void.
-PJ
An important event in the very early time line was when Chris Matthews on 18 Dec 2007 discussed the Clinton operatives spreading information that Obama was born in Indonesia and had Muslim roots. See this article for moe details on that: https://cdrkerchner.wordpress.com/2016/09/09/flashback-early-birther-chris-matthews-in-dec-2007-aired-on-msnbc-the-clinton-operatives-talk-about-obama-born-in-indonesia-and-obamas-islamic-background/