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If you really want to power something, you're better off getting a solar cell (available at Radio Shack?) or even a reasonably large photodiode. LED's have a very small area, hence the amount of solar power collected (proportional to this area) is small. LED's just were not designed to be used as a power source.
In any case, you may want to measure an i-v curve (current vs. voltage), either for the LED or a solar cell. To do that, you'll need to connect different resistance values across the output terminals, and measure the voltage. Then calculate the current for each reading from
current = voltage / resistance
What you have done so far is to measure one point on the i-v curve for your LED, using a very large resistance, the resistance of your meter (perhaps 10 M-ohm?) This reading is the open-circuit voltage.
Make sure that each point is measured in full sunlight, preferably with the device oriented toward the sun for a maximum reading (you can just hold the device in your hand, and orient it until the meter reading is maximized). At any rate, the device should be kept in a fixed orientation for all readings. If some clouds cover the sun during the measurements, wait until they have passed by before continuing.
I would guess that using resistances between a few ohms and 100k to 1M would provide a sufficiently complete plot.
You are correct that changing the amount of light hitting the device (LED or solar cel) will affect the amount of current produced. Hence the need to do a series of measurements all with the same amount of solar power.