I keep all of my image files as well as my Lightroom Catalog on a 3TB External Hard Drive and am currently considering two problems...
1) The drives are WD's 3TB USB 3 "My Book Essential" standards and ought to be identical with regards to content except for their name (working drive is called "Ellie Work," backup drive is called "Ellie BackUp"). I use a program called SuperDuper to update the backup drive periodically (the program searches for and updates any changed files rather than erasing the entire drive and recopying multiple TB of data each time; program came highly recommended by some of my IT/network management buddies). In accordance to the idea that an untested backup is not really a backup, I plugged in the backup drive rather than the work drive & tried to run Lightroom. Unsuccessful. Was told program couldn't find images. Is this because the hard drive have different names? Any ideas? Non-functioning backup is not really a backup...
2) I assumed that I'd never run out of space on such a large drive... Not so much. I take many terrible pictures while trying for those elusive keepers. How does one decide when to get rid of junk images? How bad does an image need to be to be junked? (Also, how much free space do I need on a working hard drive to allow it to continue to function smoothly?...)
It sounds like your problem is the name of the hard drive. Just to test, exit LR, rename the backup to the same name as the primary drive and be sure the primary is not connected. Then LR should go right to the back up drive and work fine.
You don't say whether you are on windows or mac. If you are on mac, I would consider getting a third drive...use Time Machine for incremental backups and do a periodic complete backup with SuperDuper. The big advantage to SuperDuper is that you can be back up running immediately as it makes a bootable backup. The advantage of Time Machine is that you can easily (even if it has to run for a while) rebuild you machine by reloading the OS fresh and installing apps and data without any possibly corrupt system problems which put you on the sideline in the first place...both are great to have in a crisis. Of course, I must sound paranoid doing 2 backups, but it's very simple once you start doing it and, yes, I am paranoid :-)
Once you've tested the backup drive, don't for get to shut everything down and rename it !