I don't know of a word or phrase which exactly describes the situation you propose. There are a variety of reasons why certain names for anything become more popular than other names, and sometimes a term does not become perceived as pejorative until some time after it is coined. While Obamacare is currently a favorite term of its opponents, The Atlantic blog notes that its origins are not so starkly negative: it may owe its popularity to newspaper headline writers trying to save space. Given the initial popularity both of the president and the bill, it might have become a positive sobriquet.
We could make a case then that what happened here is appropriation, the adoption of a word and shifting its meaning or perception. Obamacare may have started relatively neutral, but became a term of diminution because it was the name preferred by its opponents, like Reaganomics or Thatcherism. A related term we tend to hear more about is re-appropriation, where members of a stigmatized group adopt a label as a matter of pride and identity. Textbook examples are sans-culottes, queer, and redneck— the blog post suggests Congressional Democrats have no interest in reclaiming Obamacare, however.
Another sociological term is labeling, whereby some non-conforming group is assigned a label and excluded. Although not applicable to this example, it is a related concept.