How do I rename an open file in Emacs?

Yes, with dired mode, you can:

• C-x d to open dired
• RET to select directory of current file
• C-x C-j (dired-jump to the name of the current file, in Dired)
• R to rename the file (or dired-do-rename).
• q to go back to the (renamed) file buffer

The rename is equivalent to a shell mv, but it will also update any open buffers, and unlike mv it will not change the access and modify times on the file in the filesystem.

Just for completeness, since some folks may visit this page thinking they will get an answer for the "save as" feature of Emacs, that's C-x C-w for an open file.

Try this function from Steve Yegge's .emacs:

;; source: http://steve.yegge.googlepages.com/my-dot-emacs-file
(defun rename-file-and-buffer (new-name)
  "Renames both current buffer and file it's visiting to NEW-NAME."
  (interactive "sNew name: ")
  (let ((name (buffer-name))
        (filename (buffer-file-name)))
    (if (not filename)
        (message "Buffer '%s' is not visiting a file!" name)
      (if (get-buffer new-name)
          (message "A buffer named '%s' already exists!" new-name)
        (progn
          (rename-file filename new-name 1)
          (rename-buffer new-name)
          (set-visited-file-name new-name)
          (set-buffer-modified-p nil))))))

Take a look at that page, there's another really useful related function there, called "move-buffer-file".