Driving a high-power LED at very low levels *efficiently*
The amount of light LEDs put out has a very linear relationship with the current they are driven with, and the voltage to current graph is relatively constant as they are diodes with a characteristic junction voltage.  Taken together, this means that the efficiency can rise about 25% if an LED is driven at 10% of rated brightness (in this LED, but others should be fairly similar).

In any case, regardless of the level you choose to drive the LED at, what you want is a "single cell LED driver".  These are usually current-regulated, eliminating the need for any resistors which do nothing for your efficiency, and also have the ability to step-up the voltage, so you can drive a 3 V nominal LED with one or two 1.2 V NiMH cells.

Linear, Micrel, and Diodes Inc all make drivers for this purpose, and you can find many more in a search for "single cell LED driver".