XDA Forum Member avicohh released a root method for the Galaxy S III Mini, and it isn’t very difficult. For the time being, there isn’t a custom recovery available for the Mini, so users will have to flash a pre-rooted stock image via Odin.
The process is pretty simple. Users simply need to flash the image in Odin. The Galaxy S III Mini differs from its siblings because there is no key combination to launch recovery. So if users need to perform a factory reset, they have to do it in the Settings menu. To help those experiencing boot loops, the developer has provided an Odin-flashable wipe image.