You don’t copyright a name. You can register a business name, buy a domain name and get a trademark on the name. You could even get a "registered trademark" which is not the same thing. (A trademark is something you acquire over extended period of use. A registered trademark is something you can apply for even at the start of a project/business).

If your business produces any software, documents, music etc then you can copyright those works. Copyright is for protection of intellectual property such as books. Patents are for protection of innovation/ideas. Trademarks are for protection of a business/product name.

you do not copyright names – you trademark them – different animal – you must first do a trademark search to see if the name is even available – that costs money