You can rename or move your shared folders just like you would any other folder on your hard drive or via the website. Even if you rename it, the folder will still remain shared. However, changing the name of the shared folder or its location will not change its name or location in the Dropbox of other members.
This is mostly to avoid confusion by maintaining consistency among members of your shared folder. Nobody wants their folders reorganized and renamed without their knowledge. If you do need to rename or move a shared folder for all members of that folder, then you can follow the instructions below.
So why did our shared folders suddenly stop sharing? Well, you could argue that it’s because I’m an idiot, but the straight answer is this: I moved these shared folders into another shared folder, one that wasn’t shared with our internal team. This, apparently, and probably thankfully, cut out the shared-ness of the moved folders.
Bottom Line: Go ahead and move and organize your Dropbox structure. Just be clear of the permissions on your folder hierarchy.