The magnetic flux of the Earth's magnetic field over a square meter of area perpendicular to the field lines is between 0.3 and 0.6 gauss-meters^2. Let's be extremely generous and say we can harness a complete reversal of the local magnetic field for power, over the course of a year. (This is absurd, of course, but will give us some idea of the numbers we're working with.)
So the change in magnetic flux is 0.6 to 1.2 gauss-meters^2 per year, or 7.6e-13 Tm^2 per second (note: using the lower number). A Tesla (T) is a Weber (Wb) per meter squared, so this is equal to 7.6e-13 Wb/s. A Weber is a Volt-second. So, a complete reversal of the Earth's magnetic field over the course of a year would generate 7.6e-13 volts across a loop of wire enclosing a square meter.
To put that in perspective, you'd need nearly 4 trillion loops of wire to power an average (3V) flashlight.
Anyway, it was an interesting thought experiment. Slightly different numbers show that if you assume the Earth's magnetic field completely reverses over the course of one second, you get voltage of 0.6 gauss-meters^2 per second, or 6e-6 Tm^2/s. As above, a tesla-meter-squared-per-second is a volt, so you'd only need 500,000 loops of wire to power the aforementioned 3V flashlight... for the second in which the magnetic field flips. Of course, it would then have to flip back for you to get another second's worth of light.
Optimizing the amount of wire you'd need to enclose a square meter, by the way (about 3.5 meters of wire), you'd need about 1772 km of wire in order to power the flashlight for the one second in which the Earth's field flips. (Which is, by the way, about the distance from Tacoma, WA to Los Angeles.)