My iPhone 3GS has made the text message sound twice at like 3:30 am but when I look at it there is no text.
This is hard to explain to my husband, it's making us both paranoid! Can I look somewhere to figure out what's happening?
It sounds like you have another app that may be giving you untimely notifications. You can check by opening Settings.app > Notifications. Scroll through that list and tap on each application that you don't want sound notifications on. Scroll down slightly and turn Sounds off.
Unfortunately, there appears to be no (easy) way to look up notifications logs. I say easy, because notifications logs should be accessible with a jailbreak, however, as of writing, there is no jailbreak available for iOS 6 - if you have that installed.
As a last resort, if you are on iOS 6, you can also schedule Do-Not-Disturb times, from when you go to bed to when you get up. One thing you could do to verify that it is indeed the text tone, is change the text tone to something different. If the different sound goes off at night again (or anytime without a text message actually appearing to come in), that would narrow it down to it indeed being a pseudo-text-notification. At that point, Do-Not-Disturb, would seem like a reasonable "hack".
Perhaps there is a notification for when the battery is fully charged? I have that on my non-iPhone, and it typically goes off in the middle of the night if I forget to set it to silent-mode.