I bought my Mother a used iMac lampshade about 5 years ago.
It has worked flawlessly until about a month ago.
Her mouse now jumps all over the place.
I gave her several old one button mice, but they all have the same behavior.
At this age, she would prefer the one button mouse, which is what she is used to.
Any suggestions?

Depends great I guess

Mice do have problems , more so with age .

I have no problems with a Kensington pilot mouse on a Lamp Shade iMac which we currently use - well that is my grand daughter iMac , she needs the right click for them games .

Oddly enough
I thought my Pilot Mouse on my 20 iMac aluminum body was going out , since the right click did not work anymore , still does get spotty at times still . Then got a cheap Dell 2 button mouse and that had all the jumpy cursor behavior . I then repaired permissions and re-booted . The Pilot mouse works much better now .

Keep in mind that I don't shut down every night , maybe for months at a time and only runs disk utility , when something funky happens .

Do you know if it stalls when trying to move the cursor then it jumps, or does it move smooth then all of a sudden jump? Has she changed mouse pads? Does she use a mouse pad?

Did her mouse acceleration preferences maybe get bumped up?

You could tape a piece of white paper down and see if that helps.

It's smooth and then jumps.
It's an "old" optical mouse with the red glowing light on the bottom. I've tried 3 Apple mice and 2 mouse pads and same behavior.
I didn't think to look in preferences, because she would never go there. I'll check.
Could it change by accident/keyboard shortcut?
Her keyboard is set to "Sticky Keys", because she can only use one finger to navigate.

It's most likely either a corrupt kext or preference file, another app/extension conflict, or some sort of frequency interference from another wireless device nearby. Many people have run into this problem and it usually ends up being one of those things.