ROOTING GALAXY SIII NEO

Galaxy S III Neo came with the following specifications:

Display     Type     Super AMOLED capacitive touchscreen, 16M colors
Size     4.8 inches (~65.7% screen-to-body ratio)
Resolution     720 x 1280 pixels (~306 ppi pixel density)
Multitouch     Yes
Platform     OS     Android 4.3 (Jelly Bean), upgradable to 4.4.4 (KitKat)
Chipset     Qualcomm MSM8226 Snapdragon 400
CPU     Quad-core 1.2 or 1.4 GHz Cortex-A7
GPU     Adreno 305
Memory     Card slot     microSD, up to 64 GB (dedicated slot)
Internal     16 GB, 1.5 GB RAM

Also Galaxy S III Mini has the following features 

Display     Type     Super AMOLED capacitive touchscreen, 16M colors
Size     4.0 inches (~59.4% screen-to-body ratio)
Resolution     480 x 800 pixels (~233 ppi pixel density)
Multitouch     Yes
Platform     OS     Android 4.1 (Jelly Bean)
Chipset     NovaThor U8420
CPU     1.0 GHz dual-core Cortex-A9
GPU     Mali-400
Memory     Card slot     microSD, up to 32 GB (dedicated slot)
Internal     8/16 GB, 1 GB RAM

ROOTING PROCESS

   STEPS
Since Samsung phones come with a customized manufacturer interface (Touchwiz-UI)
To root the phone with third party apps may therefore be a bit hard and can Even cause Bricking
 As we hard discussed before. Rooting is getting most phone privileges in your hands and maximally utilizing them
The Previous guide was as follows
TOUCH WIZ ROM ROOT PROCESS
Use the the super one click app or z4 root

1.    Download z4 root.apk
2.     install the app by clicking the file
3.     after the file have finished installing open the app and follow the instructions
4.     to root, click the root button and  wait until the process completes
5.     after the process is complete reboot your phone
6.     Your phone is now rooted

I previously Recommended z4 root app .But due to different phone variants i highly recommend
RECOMMENDABLE METHOD
Rooting your phone using Flashable zip

REQUIRED FILES

•  Samsung Galaxy S III Neo root.zip here
• Download specific Model {Samsung Galaxy S III Mini root.zip}

Requirements:
1)Your Samsung Galaxy S3 Neo GT-I9301I should have atleast 30-40 percent of battery to perform the rooting process.
2)Root Samsung Galaxy S3 Neo GT-I9301I using Odin.

STEPS

• Download USB Driver on your computer.and install it

•  Extract the root files on your computer that you downloaded above
• Now, Switch off your Smartphone.

• After switching-off the phone, you need to boot into Download Mode.To boot into download mode Press and hold Zoom Left and Power button at same time for 5-8 seconds until download mode is active.
• In the Download mode you will be able to see a Warning Yellow Triangle Sign.In this situation you have to press the Zoom Right key to continue.
• Now, Open Odin3 (found in the extracted files, that you have downloaded in the Step 2) on your computer.
• Then connect your Smartphone to the computer.Once you have connected the Smartphone to the computer, Odin will automatically recognize the device and show “Added” message at the lower-left panel.
• Once your device is detected by Odin, click on the PDA button and select the CF-Auto-Root.tar file (you have Extracted in step 2).

• Now, click on the Start button in Odin to begin the Flashing.Flashing usually takes 30-50 seconds to complete the rooting process on your Samsung Galaxy S3 Neo GT-I9301I.
• Once Rooting process is completed you will be able to see a Green Box with Pass written on it in Odin.During this process Smartphone will re-boot automatically.
• Once you see, the Pass message in Odin, you can disconnect your Device from the computer.
• Now, to make sure you have successfully rooted your Samsung Galaxy S3 Neo GT-I9301I, open Applications Menu on your Smartphone and Find SuperSU App.