The problem is she wasn't a hired employee, she was an elected official. So you can't fire her. They could have tried to impeach her, but it would have likely failed. After all, her county in KY is conservative. In fact, she's a democrat, NOT a republican. (funny how that wasn't given a lot of press).
When a gov't duty interferes with someone's religious convictions, the state is supposed to attempt to make an accommodation. In this case, she asked that her name NOT be put on the marriage certificate. It's my understanding that initially the court would not allow for such an accommodation. Only after she was in jail for 24 hours and she was getting positive press coverage did the court reconsider.
It's my understanding the chick's been married 3 times or so, so I am a little surprised that marriage is such a sacred institution to her. Nevertheless, this is about religious liberty for the next person. Religious liberty (at least for christians) is under attack, so this is an important conversation to have.