I'm not aware of any function this serves other than letting you "lock" the release to the body when not in use, and having it stand proud for ready access when using. On most cameras of other makes that "click" like this, as you probably know, this is used for meter on/off, but as you also probably know this is handled by the collar around the shutter release on the New F-1.
All of my New F-1s, including the ones I had serviced by Ken Oikawa, do this.