The copy, rename and link dialogues accept paths with multiple / characters, e.g. /tmp//somefile or /tmp///somefile, but they don't accept multiple / characters at the beginning of the path, e.g. //tmp/somefile. To be consistent they should accept this.
The error message that is displayed is this:
Drag the icon to a directory viewer
(or enter a full pathname)
Note that the problem doesn't only affect you if you enter a double leading / in the copy, rename, or link dialogue. e.g. if you enter multiple / characters in the minibuffer it will still navigate, and when you launch a copy, rename or link dialogue the multiple / characters will be carried into it.