How to give to the creator of a node the option to hide the name of the author?
There are a few things in the original version of this question that really need review ...


  Content may add only authorized users on my site.


In Drupal you cannot have content that adds users, so I'm going to assume it's rather the opposite you're looking for, ie: "Content may only be added by authorized users on my site".


  I need to implement the functionality (option) of hiding a user name (author of сontent) if user wishes it.


Unless you have special customizations in place in your site, Drupal allows for multiple (authorized) users to author (change) content. So your "if user wish it" is a not clear enough: maybe you mean "the original author" (who first created the node), or maybe you mean "the user who last edited (changed) the content". For the sake of simplicity, I'm going to assume "the original author".


  That is, to make him/her "the anonymous author of" сontent.


If you agree with my above assumptions, here is a blueprint of how you could implement this:


Add a custom field to your content type, which is a boolean (true/false, yes/no, 0/1, whatever fits) that can be used when creating a node to have the original node author indicate if author should be shown (yes) or marked as "anonymous" (no).
Use the Rules module to ensure that the "Authored by" value always remains the "original node author" (who created the node). This basically ensures that attempts to change the node author (similar to what is suggested in codenext's answer) will be rejected. Only using Rules should be sufficient to implement this (rather basic) validation. But if needed (to further enhance these validations), add the Rules Forms Support module.
Use a preprocess function in your theme (similar to what is suggested in Zuzuesque's answer) to actually display the node author as requested by ... the original node author.


Note that if the original node author ever want to change his/her mind, that's still possible with this approach: just change the value of the custom field value as desired (after doing so, the logic in step 3. will ensure it's processed accordingly).