As for the iPhone 3G, I think it was just software and an aging device. My iPod definitely got pretty laggy with multiple apps open on a device with 128MB of RAM in an OS that doesn’t even support running apps in the background. The more mods and plugins loaded the laggier naturally.
But even with a jailbreak, they didn’t mod drivers or anything that would make it different from a hardware perspective. They just sideload a store that can then install any apps. You can install bad apps but nothing that would survive a restore in iTunes.
What could have happened is she got an iOS update after the restore that also was a bit laggier and energy intensive. Or maybe the faster discharge and higher energy consumption is what finished an already aging battery. It’s very unlikely the jailbreak caused it, more likely triggered it or expedited an existing problem. Like formatting your mom’s PC whose hard drive is on death’s bed and the IO of reinstalling an OS makes it kick the bucket. Is it the OS’s fault? No. But did installing the OS cause the fault? Yes. People will still blame the OS, especially if it’s a different OS in case of a jailbreak or putting Linux on your mom’s laptop that’s still on XP or 7. The new thing, it broke the thing!