There is something called WifiSMS, Working with 5.1.1.
This allows you to use your SMS on your desktop (text from your computer when you're at home over Wifi).
But also gives you the ability to download full conversation from someone. Once installed, go into the browser on your computer, click the conversation, and hover over the person's name (top of the conversation), and a "floppy disc" will appear for you to save that whole conversation. So far, it does not work with group conversations, but works with iMessages and Text. I hope this helps, if it does.
Examples of what the button and the text output would look like: