Try going into Settings >> My Device >> Display >> Notification panel. There's a number of icons for different functions that you can just press and drag into either be showing or hidden. Also, it could just be the Mobile data icon is still there, it's just on the other end of the Notification panel. In that case you can just slid it over to where you want.

 

Hmmmm, that is weird. The first thing I'd try to restore that Mobile data icon is tapping that 'Reset' button. (...and then you'll probably have to re-juggle the icons you want in your Notification bar.)
If that doesn't work, maybe try going into your Settings >> Application manager and then slide over the 'All' menu. Scroll down to 'Touchwiz home', select it and tap the 'Force stop' button, then the 'Clear cache' button. Restart your S3 and hopefully that icon will be back.

 

Still pondering why this happened after a Factory Reset. A Reset is supposed to fix problems not enable them?????
Try clearing the system cache partition. That's generally a recommended suggestion after any Factory Reset anyway. (probably should have brought this up earlier actually)
https://www.androidpit.com/forum/608584/how-to-clear-the-system-cache-for-the-samsung-galaxy-s4

If that doesn't help, try restarting your phone in Safe Mode. I don't think this will be a solution but at least it would eliminate the possibility some random app is responsible for your missing Mobile data button.
http://www.intomobile.com/2014/12/09/turn-safe-mode-galaxy-s4-s4-mini/

On last thing is to try toggling back and forth between 3G <> 4G/LTE networks. If you go into your Settings >> Connections >> More networks >> Mobile networks >> Network mode and pick one or the other, it should restart your S4 mini as it switches to the corresponding radio.