This is a question with two parts: a support request and a feature request.
Can the Area 51 entry be changed to match the new name?
(I spent some time trying to locate "Writing" on Area 51 only to realize that what I thought was a strangely similar site was actually the same site but with a different name. I then detoured to ask about the name discrepancy at the Writing Beta site. All of that led me back here.)
If this is something that can be done, is it something that can be automatically done for all future name changes made to all beta sites?