SMS Backup

I just found an app that someone (tfadi <- I think) wrote in the Windows App Store titled

"Windows Phone Message Backup".

The app allowed me to pull down all of my backed up SMS and MMS messages from my WP8 phone to my Windows 8.1 machine, and export individual messages or entire threads to .txt files out on my OneDrive account.

That's exactly what I needed to get to my old text messages.

I just downloaded and ran it against my SMS/MMS messages.  It worked very nicely.  It has a few rough edges, but that's the beta experience.  It's from Tafidi software.

You can get the app from here.  That link will run the Windows Store app and take you to the app.

There is NOW an app for that called contacts+message backup

Contacts + Message Backup app is a Settings app that enables you to make backup of all your contacts (in VCF format) and messages (SMS & MMS including attachments) to SD card (in XML format) for later restore to same or another Windows Phone.

Backup is stored in 'backup+restore' folder on the SD card.

Known Issue: Restoring a backup of messages from a dual SIM Windows Phone restores all messages to message store for the first SIM/slot.

Note: To use this app, please go to Settings on your Windows Phone.  This app will NOT appear in your list of installed apps.

http://www.windowsphone.com/en-us/store/app/contacts-message-backup/07f078fb-cdbb-42e8-9027-25eddccbebf2

Great stuff, thanks for testing it and replying here.  Also many thanks to Tafidi** for putting this together.

I've given this a go with very positive initial results.

To add to your post:

1.  The app cannot be installed directly onto a WP8 phone.  It's intended to be installed on, and run from the Windows 8.1 device (which is why you can't find it when looking for it using your phone's Store app).
2.  Two step authentication does not need to be disabled to use the app.  One just needs to set up an "app password" here: http://windows.microsoft.com/en-us/windows/app-passwords-two-step-verification.  Once you follow the directions on the screen you'll get a long password that you use when logging into the app on your Windows 8.1 device.
3.  The app will connect to your WP8 phone after you plug the phone into your device using the USB port on the device.
4.  The app needs to have focus in order to load all text messages.  If you start up the app, plug in the phone, and then swap to a different screen, SMS loading into the app will halt until you swap back to the app.  I didn't realise this and was wondering why the heck it was taking forever to get all my messages :P
5.  The sort order of the addressees on the left panel is not alphabetical, and not time-based (like most recent addressees first/last. etc).  
6.  Group chats are supported, but I notice the last one or two messages in the chat aren't populated.  I see the "bubbles" on the app screen, but the actual text content is missing.  Doesn't seem to happen with single-user chats.
7.  Downloading to text is easy - right-click a bubble on the right, or right-click a sender/receiver on the list on the left.  Options will appear in the lower right panel to download as text or message.  You'll be prompted for the target location and name and the export is done as expected.
8.  Downloading to image is trickier but not bad.  Only the currently-highlighted (selected) bubble is exported to image.  I was hoping i'd be able to download all images sent to me by a sender by right-clicking the sender's name on the left but that doesn't work as expected.  Still not bad and I'm very happy to have this option.

** I was originally put off by the developer's website - there's no mention of this app on the web page, and it looks a bit odd to have just a "Back to the Drawing Board" message up front.  Hope he/she gets more content on the landing page or less adventurous folks will shy away from this helpful app.

If you want to retain the texts, it is possible to copy the content of the texts and paste them to a OneNote document.  From there, you can copy/paste to wherever you like.

This is a manual process per individual text.  There's no automated way of exporting multiple text messages.