Folks at paranoid have churned out Android 6 ROM for the old Samsung Galaxy S3 i9300 phone. It is great for putting a new life into your old android phone as this ROM will give you pure android experience so no or very less lags here and there.
Download the ROM from here and flash it using CWM or TWRP of S3 i9300. Gapps may not install at first and you will make the ROM boot first > and then flash gapps. Wipe the data and then reboot again. Other than this, there are no other bugs. In case your device as difficulty in booting, try this fix.
This ROM is designed to give you better performance and stability and we are sure you will love it.
The last update Samsung rolled out was Android 4.3 that dates way back in November of 2013 which is dumping someone after an year or so. Since this S3 android phone was the best selling galaxy phone of all time, Samsung should have supported it a bit more but fortunately we have XDA to rescue us.