OP:
If I recall, Apple had "a bad run" of 3tb hard drives that went into the iMac line.
Are you going to pay for this replacement, or are they covering it?
Does your mom's iMac run at all right now?
Or, does the drive problem keep it from booting altogether?
Seems to me you're going to have to get the iMac up-and-running to "do a backup" from it.
I
highly HIGHLY second Coastal's advice above about using CarbonCopyCloner to create a cloned backup if you can. If the internal 3gb HDD portion of the fusion drive has problems resulting in corrupted data, CCC has a better chance of succeeding in "getting what's still save-able" from the old drive. (Coastal
explained why in his post above).
CCC is
FREE to download and use for 30 days (very easy to use). Get it here:
http://www.bombich.com/download.html
You'll need an external drive large enough to hold the contents of the internal fusion drive. (depends on "how much stuff" is on the internal drive)
IMPORTANT:
You must understand that Apple probably won't "give you the old drive back".
Not sure if they do this, even if the owner requests it.
But even if they do...
... you probably won't be able to "get any data" from it, because it was "part of a fusion drive". That means it was "melded to" the internal SSD portion and the data from a fusion drive may not be "recoverable" if you have only 1/2 of the drive (the HDD portion).
SO... what this means in real terms...
You've got to try to "get what you can" off of the internal fusion drive BEFORE you take it to Apple.
If the iMac won't boot, it -might- be possible to boot it from an external drive, and then access the internal drive that way (but still may not work if the HDD portion of the fusion drive has "gone bad").
If you can't "get to it" this way, because there was no backup, looks like your mom might have to "start over from scratch"...