How to back up text messages from iPhone 5S? What is the most simple way (for a non-techie) to back up my text messages, including the pictures within iMessages?
I believe that the right way on doing that is with an iTunes Back-up. This way you make sure you can restore them back to your device anytime you want without loosing any of them including the images.

Also there are few apps out there that could help you export the messages, but not sure if they can be imported back to your device. For example using an app for Mac called iExplorer.