TXT msg them (forward them) to an email address instead of a phone number. That's what I have to do. In my case, that's what happens when your phone is 5 years old -- and I don't think it was newest or never even then. Outside of forwarding messages, I'm really not aware of any other way to "get them off" of a phone.
I've never tried the free service http://www.treasuremytext.com
that claims to be able to save text messages.
That's mostly because it's a UK service, and would be long distance from North America (USA, Canada).
iPhone users can connect their phone to iTunes, and use a paid app like SMS Export. I don't know about jailbroken phones, however. I'm like you -- once it's hacked and it works, I'm a bit skittish about messing it up, because that can happen quite easily.
Even if you somehow find a way to connect your mom's phone to yours (and I don't see how you could), there's no real way to move the messages across.
Again, forwarding to an email address instead of a cell phone # is probably the best way.