Assuming the server is Linux, you can easily use 'autofs' to automatically mount your USB drives at a predetermined location. Your backup script will need to be smart enough to look at slightly different names for each USB drive's mount location. But those locations are predictable.
The bonus of 'autofs' is that it only mounts the drive when needed, then will dismount the volume after a configurable period of inactivity.
Create/Edit /etc/auto.master and insert where you want to mount your USB drives, the configuration file that maps each drive to the location, and the dismount timeout period.
# USB backup drives get mounted at /mnt/offsite/DRIVENAME
/mnt/offsite /etc/auto.offsite --timeout=1800
Create/Edit the /etc/auto.offsite file.
# Each drive needs to be formatted with ext4, and UUID determined
OFFSITE1 -fstype=auto,rw,noatime,data=journal,commit=1 :/dev/disk/by-uuid/b5c1db0d-776f-499b-b4f2-ac53ec3bf0ef
When the USB drive with that UUID is attached and you access the directory '/mnt/offsite/OFFSITE1', it will auto-mount and be ready for access.
Other then having to figure out the UUID of the USB drive after you format it with ext4, editing the auto.offsite file, and then restarting the autofs service; this results in a very foolproof method for hooking up external USB drives. The users just need to know enough to plug / unplug the drive each day/week. Everything else can be handled via scripts (checking the drive mounted properly, checking that the backup ran, checking free space, etc).