Can I rename or move a shared folder?
Yes, but it will only rename for you. If you want to rename it for everyone you need to unshare it, rename it, and re-share it. Here is a dropbox article explaining it.
However, in IAM you can ask a dropbox administrator to rename it for you. We have the ability to rename any shared folder so that the change is effective for all dropbox accounts registered with the IAM dropbox system. This does mean that if someone hasn’t registered with the IAM dropbox system they would see the folder with its original name still. Also, whenever you share the folder with someone, it will be shared with them with its original name and then overnight it will be renamed to the current name. A small price to pay for convenience…