I want to know what is the best standard way to mention a website developer's name. I mean is there specific meta tags? In case that customer does not want the designer ads its name in the footer or somewhere in the vieable content.
Here are the valid name attribute values:
There doesn't seem to be one specifically for the site developer, but that doesn't mean you can't put it somewhere in a generic meta tag, or define your own name value.
Really, though, if you care about this you should have agreed upon it with your client before signing the contract. Since it's advertising, some developers may decide to charge a lower rate if they mention their company in the footer or a higher rate if the client wants it completely white-label.