Why do I get a 'connection to the outgoing server "smtp.gmail.com" failed' message in iOS Mail?
I occasionally get a call from a client of mine complaining about this.  I don't know of an actual fix (or what the actual problem is) but I usually recommend doing these in order:


Quit Mail fully: double-tap the home button, swipe up on Mail
Normal restart: press and hold power until you get the option to shut off, wait 30s for it to shut off, press and hold power until
the Apple logo shows up again.
Try again and if it fails, forced restart: press and hold power AND the home button until it shuts off - keep holding until the
apple logo shows up again
Remove and re-add the gmail account.