Are you certain that the screen stays black? (no image at all)
Make sure by trying to boot. Let it sit for a couple of minutes, so it does try to boot, and --- if the video was working properly, you would expect to see (something/anything) on the screen. After that wait, shine a bright light into the screen from several angles. If it needs to log in, you may see something that looks like a login screen in the center, even if the image is very dim (because the screen backlight is not powered on to provide normal brightness to the screen) If the Mac is set to login automatically, you should be able to see (something) at the top of the screen/menubar. You have to shine your light into the corners, looking for the Apple icon in the top left corner, or the clock near the top right corner, or the dock/icons along the bottom edge of the screen, or ANY other icons/files, or any other screen elements. Those may be difficult to see, but you need to rule that out.
(If you can see dim items, then there's other parts that might be replaceable, not the LCD screen itself. That will then depend on which iMac model you have)
just my 2 cents: Sounds like your relative was taken for a ride by that repair shop. I hope they didn't have to pay much for the display replacement.
It can help if you can determine which iMac this is... You can find the serial number for this iMac on a label on the bottom of the stand. You can then search for "Apple serial numbers", where you will see several sites that will decipher the serial number, telling you the exact model, and other tech specs for that iMac.