Basically, I want to make a backup of my text messages from my phone to my computer (so I can free up the phone space and keep a record of past conversations). The backup can be in any format (plain text, cvs, html) as long as it is viewable on a regular computer. I want to get the name, date and content of the message (which will be text and/or a photo (not critical if text only). I have thousands of texts. The solution must be free. If it requires a app on the phone (which I'm assuming it will) the app must be less than 8mb.
I can highly recommend SMS2Text.
It meets or exceeds all your requirements: It will backup your texts, will allow them to be viewable on a regular computer, will export the name, date, and content of each message, and is free. It will work fine on thousands of texts.
You asked for it to be under 8MB... SMS2Text is currently under 1MB!