What should I do if someone is claiming copyright infringement of my business name?

Copyright protection is directed to protecting content. Trademark protection is directed to protecting brand identity. Typically a business name would be protected by trademark law. In order for a phrase to function as a trademark, it must be capable of identifying the source or origin of the goods or services it is associated with. If a term describes the goods or services, it cannot function as a trademark or service mark. Typically terms like "Bookstore", "Carwash" and "babysitting services" would not be likely to function as a trademark or service mark. Each alleged trademark term and usage has to be evaluated in the context of the surrounding facts and circumstances. I recommend discussing this with your trademark attorney to advise you about the particulars in your situation.