Resolution:
This issue is only with the folder resource. All the other resources under this folder are not affected.
This is a defect in Designer 4.1.x and there is a workaround suggested to resolve the issue.

Workaround:
User needs to build the folder structure as he desires in the new project and then do Import resource with the "option to overwrite set to off" (by default, this is set to off).

1.Say the Existing Project has :
+Folder1  ---&gtExport at this folder level and create a testexport.xml file
  +Folder2
     *Process Definition

2.Open the new project newimport.dat and
    Create Folder1 and Folder2 as below: Note the Folder1 and Folder2 has the "Rename" option enabled.
     +Folder1
         +Folder2

3.Now Import Resource with " OverWrite in case of Conflict" box unchecked (by default it is unchecked) under "Edit Advanced Settings". This way, Folder1 and Folder2 will not lose its "Rename" mode.

4.Now the resources under Folder  would be correctly imported and all the folders have the rename option enabled.

This issue will be fixed in the release of Designer 5.0.

When youexport the folder resource and then import this to a new project, you won't be able to rename the folder anymore. If you right click on the imported folder, you will find "the Rename" option disabled.