Mine doesn't rattle. If it did, I'd use basic troubleshooting and repair methods. The rattle is vibration from pieces that are supposed to be tight, but are loose for some reason.
First determine where the rattle is coming from. There are two ball joints and the mount. And there is some linkage inside the mirror. If you don't know where the rattle is coming from, touch the ball joints, the mount, and the day/night adjuster and see what makes the rattle stop.
If the rattle is at the base and mount and the screw doesn't tighten it, it was probably overtightend at some point and the base is bent. You might try to bend it back out or you might want to try to use some tape as shims for a tighter fit. Remember, you want a nice snug fit without having to use the screw. Then use the screw to hold it secure to the base.