The best iPhone app that will save text messages is considered to be an app by Decipher Tools called Decipher TextMessage. The app allows you to save and even recover deleted text message from iTunes backups. You can also save and view MMS images and videos. It has a search function so you can easily find a specific text if you want to and will allow you to export plain text conversations that will give the full history, images and any attachments that were included.

The Decipher TextMessage app costs $19.99 with a thirty day money back guarantee. There is a free trial available so you can try before you buy. Along with being a great iPhone app to save text messages, it also works with all your other Apple devices.