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Very high efficiency GaAs light-emitting diodes are based on surface-textured thin film structures. The technique relies on surface texturing by "natural lithography", where a monolayer of randomly positioned polystyrene spheres acts as a mask for etching a random diffraction grating. We present results of a systematic experimental study on the influence of the surface-texturing parameters on the efficiency of these LEDs. The study was performed on GaAs/A1GaAs structures optimized for photoluminescence and electroluminescence, respectively. It shows that the maximum enhancement of the light output occurs for spheres of 200 nm to 300 nm diameter, which must cover more than 50% of the surface. The optimum etching depth is approximately 160 nm. Using these conditions, an external quantum efficiency (corrected for the respective emission wavelength) for MBE-grown GaAs light emitting diodes of 10% was achieved for a device of only 50x75 .tm2 in size.