I found a working solution at last!
First, create a group and add the user that needs access to the GPIOs to it (this is for the current user;
replace $USER with an explicit username if you want).
sudo groupadd gpio
sudo usermod -a -G gpio $USER
Second, add the following lines to /etc/rc.local (you will have to edit it as root, using sudo vi or whatever).
chown -R root:gpio /sys/class/gpio
chmod -R ug+rw /sys/class/gpio
Third, create a new udev rule in a file, say /etc/udev/rules.d/80-gpio-noroot.rules (you will have to edit it as root again):
# Corrects sys GPIO permissions on the Joule so non-root users in the gpio group can manipulate bits
# Change group to gpio
SUBSYSTEM=="gpio", PROGRAM="/bin/sh -c '/bin/chown -R root:gpio /sys/devices/platform/INT34D1:*/gpio'"
# Change user permissions to ensure user and group have read/write permissions
SUBSYSTEM=="gpio", PROGRAM="/bin/sh -c '/bin/chmod -R ug+rw /sys/devices/platform/INT34D1:*/gpio'"
Fourth, reboot or restart udev using
sudo udevadm trigger --subsystem-match=gpio
And finally test:
mraa-gpio get 16
The first call will not do anything but set up the GPIO, so the value will be -1, but following calls should work. And return a value between 0-1.