Rooting will allow you to install applications that require super user access (Super SU). You will also gain access to system files that can modified on the phone itself. To root your Galaxy S3 mini, follow the steps given below.
1) Download this zip file and place it on your sd card.
2) Reboot into recovery on your Galaxy S3 Mini.
3) While in recovery, choose ‘install zip from sd card’.
4) Now, choose ‘choose zip from external sd card’ or ‘choose zip from internal sd card’ depending on where you kept the root.zip file.
5) Select the root file and flash it using the power button.
That is it. You have successfully rooted your Samsung Galaxy S3 Mini. Keep in mind that rooting voids the warranty of your cell phone.