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What's in a name? A lot, with the name 'Isis'

Like most people in the United States, David Emami, an Iranian American who lives in Oregon, has a lot of problems with the Islamic State in Iraq and Syria, also known as ISIS.

"The whole situation is horrible on so many fronts," said the 31-year-old. "We have a terrorist group that has hijacked an entire religion, tarnished a name, and they are as barbaric a group as we have ever seen."

Militant Islamist fighters parade on military vehicles along the streets of northern Raqqa province, June 30, 2014.

He has one other concern: "What they are doing and have done to my daughter's name is truly heart wrenching."

Emami's 3-year-old daughter is named Isis. "We named her after the Egyptian goddess," he said. "She's a beautiful little girl who is proud of who she is, and she loves her name!" While Emami said no one has teased his daughter about her name yet, "I've experienced several beheading jokes and heard—'Better not ground your daughter, she might behead you.'"

President Barack Obama and his administration prefer to call the violent group spreading across Iraq and Syria "ISIL," which stands for the Islamic State of Iraq and the Levant. However, most media outlets continue to refer to the group as "ISIS" for Islamic State in Iraq and Syria.

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Last week, Isis Mobile Wallet changed its name to Softcard in response to the branding dilemma. A beauty magazine in the U.K. called ISIS Mag has reportedly changed its logo to emphasize word "Mag."

"We have had internal discussions," said Amy Blackley of Isis Pharmaceuticals, a 25-year-old company in Carlsbad, California. The company's ticker symbol is, simply, ISIS.

"It's unfortunate," Blackley said, but after discussing whether to change the company's name, Isis Pharmaceuticals decided to "ride it out...we are not a retail-driven company."

Blackley said she hasn't heard of any concerns from investors or members of the medical science community about the company's name. "We've seen very little impact on our business," she said.

But it's been too much for Isis Martinez, who was named after her mother.

"I had become overwhelmed by TV monitors in various public places continuously displaying and continuously saying my name preceded by such despicable adjectives," said the founder of a holistic health fund. In late August, she posted a video on Youtube and has started a petition to demand the media refer to the terrorist organization as ISIL.

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"Accuracy in journalism is not an option, it's a responsibility," she said in the video. In an email, Martinez said, "The straw that broke the camel's back was two weeks ago when I went to the emergency room to be treated, and an intake nurse who was seeing me was reluctant to say 'Isis' when she first met me."

She said the nurse suggested that Martinez use her middle name. "I know she didn't mean any harm by it, but I felt terrible," Martinez said.

Comments on Martinez's online petition include one from a Dr. Isis Perpetuo: "I live in Brazil, and this is (affecting) me here. Please, change the name for ISIL, please."

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Emami said there are more than 3,000 women named Isis in the U.S., and he supports Martinez's petition. Type in "Isis" on Linkedin, and more than 9,600 results come back.

"I can't imagine telling my 3-year-old that she needs to use her middle name, or that she's no longer Isis," Emami said, though said he might consider a name change if he feels his daughter's safety is in jeopardy. "I've had people say I should have another kid and name him Charles Manson—all kinds of horrible jokes."

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Blackley, at Isis Pharmaceuticals, said the only changes to "business as normal" that she's seen is a more cautious approach to the public. "I'm more careful answering the phone, 'This is Amy from Isis.'"

She added that the company is more careful about travel plans. "You don't want a driver showing up at the airport holding up a card saying, 'ISIS.'"

—By CNBC's Jane Wells