How can someone remove their name from an internet site.

Yes, you could sue the website under this set of circumstances or more realistically try to file a complaint with their web host for violation of their Terms of Service.

My take is that many more facts are needed before it's possible to conclude that anything unlawful has occurred.

The gist of your complaint is that your sister's name is listed "on a derogatory website." I don't know what that means. Websites just are -- they are agnostic and are not "derogatory." What could be derogatory (by which I think you mean defamatory) is the content on the website that relates to your sister. Without knowing what is being stated or implied about your sister no one can tell you if she has a defamation claim against the person who posted the information or against the company that hosts the website.

I caution you against posting any specific information about the offending website in a public forum such as this one. If your sister believes she's been harmed by the publication of her name on the website (not just angered or embarrassed) then she needs to discuss the matter with an attorney licensed in her state who practices internet law.

Presumably, your sister is not a minor. Depending on the context and content, the conduct you describe could be defamation, and it could be a variant of invasion of privacy. Much depends on the law of the state where your sister lives. A strong letter from a credible lawyer should do the trick; otherwise, your sister will have to resort to litigation against the web site operator (registrant) and the person who paid for publication of the listing.