Why is the coiled white wire on a GFCI circuit breaker smaller than the branch circuit neutral?
The size of wire required to carry a certain amount of current is primarily affected by two things:


How much power one can afford to waste in the form of voltage drop per unit length of wire.  The greater the power one can afford to waste per unit length, the smaller a wire one can use by this criterion.
How much power one can dissipate per unit length without the wire getting unacceptably hot.  The more heat can be carried away, or the greater the acceptable temperature rise, the smaller a wire one can use by this criterion.


Normal current-carrying figures assume a conductor wrapped in a protective outer jacket and then buried in material that is thermally non-conductive but flammable.  A wire in free air within a breaker box would be able to get rid of heat far more effectively than one in an insulated wall, and less likely to ignite anything nearby.  Further, if a wire is short, the allowable voltage drop per unit length will generally be much higher than it would with a longer wire.  Thus, the normal factors which limit allowable currents as a function of size, or compel minimum wire sizes as a function of current, are less applicable for short wires in a box than they would be for longer wires in a wall.