You probably need to be running Raspbian to use RPi.GPIO as a non-root user, possibly it needs to be jessie as well.
Does /dev/gpiomem exist? If so RPi.GPIO should use this device to access the GPIO. If /dev/gpiomem does not exist try:
sudo rpi-update
to make sure you are using a recent kernel, and
sudo apt-get update
sudo apt-get upgrade
to make sure you are using a recent RPi.GPIO.
You will also need to make sure you are a member of the gpio group:
sudo adduser pi gpio
EDITED TO ADD
The Raspbian system appears to be in a state of flux. I'm not sure if it's a Raspbian or (more likely) a raspberrypi.org problem.
Check that /dev/gpiomem has the correct permissions.
$ ls -l /dev/gpiomem
crw-rw---- 1 root gpio 244, 0 Dec 28 22:51 /dev/gpiomem
If it doesn't then set the correct permissions as follows
sudo chown root.gpio /dev/gpiomem
sudo chmod g+rw /dev/gpiomem
