Messages not working on iPhone 5
Press the Home button twice. This will bring up recently used apps. Find the Message app and swipe up. This will force close the app.
Hold down the Power button until the power-slider appears, but don't turn the device off. Instead, while that power-slider is showing, hold down the Home button until your home screen re-appears.
Hold down the Home and Power buttons until the Apple Logo shows, and then release. 
If one of your recent messages had a photo or video, try deleting it from the Messages. This may keep it from crashing again if you're able to open the app at all.


→ Try this multiple times.

If these steps still don't fix the problem, try backing up your iPhone to iTunes. Then restore from the backup. This step is tedious, I understand. But it may solve the problem when everything else has failed.

EDIT: 
(additional suggestions)

Going beyond what usually fixes this problem...

You're options, shy of going to an Apple Store for an actual diagnostic, would be limited at this point. I see two options.

• Restore as a new iPhone.

• Try downloading PhoneClean on your computer. I have only used this briefly and am not entirely proficient in this app, but it has helped me in the past. It's available for Mac and PC.