Mom’s imac is misbehaving. I think it’s a hardware failure and likely means it’s a dead duck.
I am using TeamViewer to see her screen so this screenshot may not be exactly what she sees but it matches what she describes. I am thinking I will have her ship it to me (ugh) but before I do, wanted to see if anyone has an easier idea. It’s an old 27" iMac 2.16 GHz Intel Core Duo model iMac6,1 (2007?) running OS X 10.6.8.
Not sure about that model but I have replaced a few video boards in later (c2011-2012) models. Not a hard DIY to do at all. Last time I did one I bought the board from Amazon - seems like it was about $80. The only tricky part is handling a very fragile ribbon cable that connects to the monitor. Other than that it’s a few screws and a few connectors.
Looks like the graphics card is gone.
Time for a new iMac 
Deader than Elvis, baby! But hey, Christmas is coming soon so there’s your excuse to get her a new Mac!
10.6.8 means no security updates, I believe. I’d take that into consideration and favor a new machine.
Or, get yourself a new Mac and give her your old one.
MacTracker says she can update that to 10.7.5, by the way.