Question about Galaxy S3 Mini: Weird Problem
There's a few options that I can think of.


Reinstall your ROM. Do it through Odin on PC or Heimdall on *NIX systems. This won't void your warranty, and will just install the ROM again. Ensure it is EXACTLY the correct one for your device / network. Check Sammobile for your firmware.
Extract the keyboard. Download your firmware zip and extract the keyboard.apk and install it manually.
Find the apk online (HUGE security risk IMHO) 
Use an alternative keyboard. 
Try wiping your cache in recovery and rebooting see if it fixes the issue. 
Check if the keyboard is there at all in your 'all apps' list and force start, clear data, reboot.  (There probably isn't, but look for Google Keyboard as well, no harm)