The Atlantic has done some digging and now reports that it's found the answer. In March 2007, lobbyist Jeanne Schulte Scott, in her analysis of the 2008 presidential campaign, wrote this in the trade magazine Healthcare Financial Management:
The many would-be candidates for president in 2008 are falling over themselves offering their own proposals. We will soon see a "Giuliani-care" and "Obama-care" to go along with "McCain-care," "Edwards-care," and a totally revamped and remodeled "Hillary-care" from the 1990s.
The word is most often used as derogatory shorthand for the Affordable Care Act, passed in March 2010, which was billed by the Obama administration and its supporters as a way to cut costs and provide health care to millions and derided by opponents as an expensive, job-killing government takeover.
Now the term has become so commonplace, it's even spawned a spinoff, "Obamneycare", which has been used repeatedly to attack Republican presidential candidate Mitt Romney. The former governor of Massachusetts championed a health care bill in his state that, President Barack Obama's top adviser has said, provided much of the framework for the Affordable Care Act.
It should be noted that the person often credited for coining Obamneycare, former presidential candidate Tim Pawlenty, has endorsed his former rival.
The Washington Post reports that health care reform advocates are looking to put a positive spin on the word Obamacare, using the term in their campaign to tout the benefits of the Affordable Care Act.