Charger / USB cable variability and problems
The forums are all filled with comments from (more or less experienced) people complaining that recharging their S3 might be at times strange with a certain ROM or another - and while maybe 10% of the time that might be indeed the case, in 90% of the cases it boils down to problems with CHARGERS and USB CABLES.
Right now it seems that the amount of power used for recharging is controlled primarily by S3 itself using at least 3 levels - of which at least two can be fine-tuned in software (and some kernels expose those features).
There is an "AC charger level" (set around 1000 mA), an "USB level" (set around 500 mA since USB1/2 standard does specify that) and a 3rd "low level / safety level" which seems to be around 50-100 mA.
The S3 seems to be choosing the level based on BOTH the charger and the USB cable used - and I have seen the same charger showing a different level with different USB cables, and the same USB cable showing a different level with different chargers. I have even seen one AC charger that with a very specific cable (not the one bundled with it) was showing "charging/USB" inside the S3, while all other cables on the same charger were showing "charging A/C".
So when your S3 charges very-very slowly your very first check should be with another USB cable, and second check should be from a different charger or USB port - instead of posting same questions again and again around here
IMHO the absolute safest charging combination right now is with the original Samsung USB cable from the USB port
(ideally in a good notebook - good notebooks have very effective USB protection, and some - like the Thinkpad models - have an USB port that can power something even when the notebook is stopped, and with one extra BIOS setting even when the notebook is stopped and not connected to A/C)! This is not the fastest charging combination
- and with an extended battery it might take like 9 hours for a full-full charge (4 hours with normal battery) - but is most likely the safest (simple) one! Also by charging with a smaller current the heat generated is (much) smaller and the "aging" of the battery is drastically reduced.
If you are in a huge-huge hurry you might want to use the original Samsung A/C charger plus the original Samsung USB cable, but don't "overdo it" by trying forced/exotic solutions/chargers - we even have an entire thread around here at XDA with "sudden death" cases - most of them observed in the morning after the i9300 was left overnight in some A/C charger
Also if you see a very-very slow charge (or even a discharge) while you are "charging" but you have the screen ON and the CPUs running at full speed you should know that is possibly normal - since screen ON + CPU at 100% is draining more than 500 mA (= probably most of, or more than you are getting from the cable); try to lock the phone or even power it off, and then after 20 minutes see if the battery is better and by how much.
As a far later step if everything else fails you should also check if the pins inside the USB connector in the S3 are not bent in any way, but that is slightly more advanced and might not be perfectly fixed without replacing the connector.
Also if your phone has other serious hardware problems (cracked screen or so) you should mention that first.
My stuff above was "educated guess" - now we also have 'hard technical data":