Move a VirtualBox Virtual Disk
It's actually embarrassingly easy, really.
Go to File > Virtual Media Manager and select the disk you want to move. Make sure the machine using it is offline, and press Release. This will disassociate the disk form the machine. Then chose Remove and opt to keep the disk image. That's important, because otherwise you'll lose your data!
Proceed and move the disk file to the new location.
Open settings for the virtual machine using the disk, go to Storage, select Controller: SATA and tap the Add button below the list and chose Hard Disk. Opt to pick an existing disk and find yours using the file browser.
VirtualBox doesn't allow you to import disks already mounted or already tracked by the platform. Every disk has a unique ID which identifies it, which means that simply copying and trying to attach the copy won't work at all.