First download and install this Chinese universal rooting program from Here
Then
1- Turn on USB debugging in your Galaxy S3 mini's settings
2- Connect your phone
Wait
NOW YOU ARE SUPPOSED TO BE DONE
Check if superuser app is now installed on your phone.
If you don't like the Chinese Super User app,
go to playstore and instal SuperSU or SuperUser
Then update it's binary, the uninstal the Chinese Super user app using root uninstaller or
any root file manager you want