How do you save old voicemails from an old iPhone?

You could record them onto another medium—a tape recorder or a CD.

One way of doing it would be to use a 1/8 inch to 1/8th inch cord. You could just run the cord into the audio input on your computer and use something like Audacity to record them.

I found these options:
http://www.macroplant.com/iexplorer/tutorials/how-to-access-voicemail-on-iphone
http://www.instructables.com/id/How-to-Download-Voicemail-from-an-iPhone/
http://www.howtogeek.com/141164/how-to-backup-iphone-voicemails-to-mp3/

I have some of my Mothers saved messages, they make me smile when I hear her.

If the phone has an SD card in it, see if the phone will allow you to move it to the card. Take the card from the phone, and insert it in the card reader on your laptop. Open Finder and see if you see the card and it’s files. If it does, you should be able to move them to the computer or make a copy.

perhaps you could forward the voicemails to yourself and save them on the new phone.