The SGS3 Mini that I'm rooting is currently running Android 4.1.2 (build I8190XXAMA2) stock firmware.
If you have a model other than i8190 such as i8190N, or 8190L, check out the "Root and Install CWM Recovery on Samsung Galaxy S3 Mini GT-I8190, GT-I8190N & GT-8190L" tutorial over at DroidViews.
I like this method as it does NOT wipe your phone, so no data is lost. Backing things up is completely optional, but still recommended in case things go belly up.
Make sure your phone is fully charged. While you're waiting, download the required files below.
- Grab a copy of the pre-rooted image (I8190XXAMA2_rooted.zip).
- Download Odin3_v3.04.zip
- If needed, make sure your phone is detected properly on your computer with Samsung's USB driver for mobile phones
Rooting good time
- Extract the zip archives.
- Run Odin3 v3.04.exe
- Click the "PDA" button and select "I8190XXAMA2_rooted.tar.md5" you extracted from the other archive
- Turn off the phone
- Put it into download mode by holding Volume Down + Home + Power until it turns on
- Plug in your phone and wait for the "ID:COM" field to turn blue
- Take a quick look to make sure "Re-partition" is NOT ticked
- Click on "Start" and wait for it to finish.
- It phone should restart on it's own once ready.
- Allow it time to do what it needs to do.
- Once you're back into the phone, the superuser app should be in your app drawer!
Images are from Tyxerakias on XDA