If you want to edit the raw data, you’ll need to be very careful as there will be references to the blog scattered all over the place. Please do this on a copy of the site to be safe. Search and replace on all possible variants of the blog’s full URL and slug (or subdomain) should do the trick – however, be aware that if the blog’s slug/subdomain isn’t unique to it and it alone, then global search/replace could mess other items up. Since the blog ID wouldn’t be changing, most programmed linkages and assets should be unaffected. Proceed with caution and do lots of searching before you do any replacing. Did I mention that you should do this on a copy of the site to be safe?
The simplest (and probably most reliable) way would be to export the group blog content, unlink the existing group blog from the group, create a new group blog, then re-import the content. This wouldn’t recreate the activity items in the group, but is the least technical method for renaming that I can think of. It might then be possible to search-and-replace on just the BuddyPress activity items table to point legacy items to the new blog.
Either way, I’d do this on a copy of the site first.