1. Get the following files from a Windows 10 1607 installation:
C:\Windows\System32\syncui.dll
C:\Windows\System32\en-US\syncui.dll.mui
C:\Windows\System32\en-US\shell32.dll.mui
(Replace en-US with your locale code)
2. Copy syncui.dll and syncui.dll.mui to their respective locations in System32.
3. Make a backup of shell32.dll.mui.
4. With Resource Hacker, add the missing text strings for Briefcases from Win10 1607 shell32.dll.mui to the current system's shell32.dll.mui.
Copied from Win10 1607 shell32.dll.mui to current system shell32.dll.mui:
Missing strings from String Table 298
Missing strings from String Table 330
Missing strings from String Table 405
Missing strings from String Table 406
5. Export HKCR\Briefcase and HKCR\CLSID\{85BBD920-42A0-1069-A2E4-08002B30309D} from Win10 1607 and import it to the current system.
Briefcases imported from older Windows versions should show as Briefcases with their icon. I could not figure out how to get it to show in the New context menu. I have not tested whether syncronization works.