Intro

My SGS3 mini is pretty good phone for its price. After seeing phones with plain Jelly Bean OS I started to see if I could get rid of TouchWiz. I did not find any good instructions how to do it especially without Windows computer. I was not particularly interested in bricking the phone so hesitating took more time than actual steps. If somebody takes any advice  from this writing it's possible that you have precious new paperweight and I will not take any responsibility of it. I do not have any idea what's the status of warranty after these steps.

Prerequisites

Here is list of essentials I used:
- host computer: laptop running RH6.4
- usb cable that works (they all do not!)
- SGS3 mini of course :)

- heimdall, installs  to host computer

• this one is used for flashing and potentially quite dangerous tool I guess
• installation complained something about version mismatch with libusb1 but I used --nodeps with rpm command
• installed also frontend but I did not use it

- various packages:

• recovery-clockwork-touch-6.0.2.7-golden.tar
• this one is recovery tool to be used for various tasks like backup and OS
• unpacked it, it contains recovery.img
• cm10.1_golden.maclaw.20130908.zip
• Android 4.2.2
• gapps-jb-20130812-signed.zip
• I also installed sshd server to phone for later use

Installation

First I charged battery of the phone. I backed up essential stuff but fortunately did not have much. Next I booted up the phone into download mode. This was done by shutting down the phone and starting it up by pressing down <power>, <vol down> and <home> -buttons. Once some warning are being issued on the display I pressed <vol up> -button. Then I connected to host computer via USB cable. If everything is fine

'heimdall detect' should announce that it has detected phone.

Next I had to find where to flash the image: I issued command 'heimdall print-pit' and searched the entry whose Flash Filename is recovery.img. For me it looks like:

--- Entry #20 ---

Binary Type: 0 (AP)

Device Type: 2 (MMC)

Identifier: 19

Attributes: 5 (Read/Write)

Update Attributes: 1 (FOTA)

Partition Block Size/Offset: 491520

Partition Block Count: 32768

File Offset (Obsolete): 0

File Size (Obsolete): 0

Partition Name: Kernel2

Flash Filename: recovery.img

FOTA Filename: 

Then bravely into flashing: 'heimdall flash --Kernel2 recovery.img'. This passed successfully and the phone rebooted. 

After reboot everything seemed to be fine  I started sshd -server and transferred cm10.1_golden.maclaw.20130908.zip and gapps-jb-20130812-signed.zip to external SD card on the phone.

Then I shut down the phone and started it up this time pressing down <power>, <vol UP> and <home> buttons. I got  into clockworkmod recovery -mode. I took the backup to external SD card and then wiped data/factory reset, wiped cache and wiped dalvik cache.  Next I  installed  cm10.1....tar.gz package from external SD card and after this gapps...tar.gz the same way.

That's it, folks! After reboot it showed cyanogenmod animation and started up. So far I have not noticed any problems at all. Actually there is one thing missing:
FM  tuner is not in CM10.1 - bought Spirit FM Unlocked from google play and it really rocks.