Install Facebook Messenger on iPhone 3Gs running iOS 6.1.6
Solution 1:
It will let you download the last compatible version if you already have the app in your purchase history. You can add the app to your purchase history using iTunes on your computer, since iTunes is version-agnostic.
Once you've downloaded the app on iTunes, it will be in your purchase history and attempting to download it again on your device should prompt you to accept an older version of the app.