You can install Handcent from the Play Store, that is the messaging app I use, and it has the ability to turn on reminder notifications. Just make sure to turn off notifications in the stock messaging app so you don't get double notified.
I am looking for a app that will notify me if I do not open a text message.
Which phone do you have? Most of the phones will have the default notification for text messages that will appear at the top of the screen, or actually change the text icon to indicate new unread text messages.
I have the HTC and yes it'll ring one time when a text comes it and blink until I read it...but I don't look at my phone unless I'm going to use it so I want app to ring until I read it