Do I need to include my real name in copyright notices?

The use of a copyright notice is no longer required under U. S. law, but it's still often beneficial since it gives notice that there's a rights claimant. The name used on a copyright notice is supposed to be "the name of the owner of copyright in the work, or an abbreviation by which the name can be recognized, or a
generally known alternative designation of the owner." Your domain name might qualify as an a.k.a., "also known as" "generally known alternative designation," though then your copyright notice will depend on you also maintaining this domain name.

Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.

Notice: The below is for educational and informational purposes only, is not a reflection on or a representation of any views or opinions held by my employer, and should not be construed as legal advice. Nothing herein is meant to create an attorney-client relationship.

Regardless of the name you choose to list in your copyright notice, you would still need to register the copyright in your work in order to sue for infringement in the U.S. The question then becomes, who is the author of the work and who is the actual copyright claimant (if different from the author)? I suspect the answer to both questions would be one and the same, i.e. you. If you are a sole proprietor with an a/k/a or d/b/a, you might be able to sue under such name. However, it would probably be best to use your own name in the copyright application. Or, better yet, incorporate or create some other form of limited liability entity in order to protect yourself fromo lawsuits involving use of your website (and any products/services you offer therein), and then file the copyright under that corporate name.

You might wish to speak to a corporate attorney licensed to practice law in New Jersey, as well as someone experienced in copyright matters.