What to do with an unstable motherboard/CPU?

Hi folks,

I've got an iPhone 4S that I originally brought into a repair shop because it wasn't charging. Thinking the battery was the problem, they replaced the battery, but the problem persisted. I brought the phone back to the shop and they determined the motherboard/CPU was unstable and they told me they risk ruining (and/or erasing) the phone if they try to fix the problem. Not having a backup, I brought the phone home and managed to back it up one last time before it died, but I did not have time to erase/factory reset it.

So, I'd like to see if I can fix the poor guy and/or, more to the point, what the problem might be. What should I do?

If it's not worth fixing, how do you suggest I dispose of it properly? I'm fairly certain all my data is still on the phone, albeit inaccessible for the time being.

I don't know a lot about tech, but I'm fairly sharp. Still, ELI5 if you can and we'll work from there.

Thanks a bunch.

- Katelyn