About 0.79% of the light gets through on average, but there is variability across wavelengths and individuals. (variability between individuals seems to be correlated with density of macromolecules in the eyelid).
This is called "eyelid spectral transmittance".
Lucky for you, Bierman et al (2011) sought to determine the answer to this question.
Using a custom LED wand that actually emitted light under the eyelid for a sensor on the outside to read, they found quite a bit of variability both across wavelengths as well as across 27 test subjects.
They report their findings in terms of optical density, which is explained as follows:
A measure of the transmittance through an optical medium. Optical density equals the log to the base 10 of the reciprocal of the transmittance.... In spectroscopy, optical density is the measure of absorbance, and is defined as the ratio of the intensity of light falling upon a material and the intensity transmitted. Abbreviation OD.
We can use the equation A = -log10(1/T) to convert their findings of OD to transmittance and then multiply by 100 to get a percentage of light that can pass through.
- Their result of an OD of 2.1 ± 0.3 SD = a transmittance of 0.79%
They do note that
eyelid thickness, and/or the density of macromolecules in the eyelid account for most of the variation in transmittance between subjects.
Bierman, A., Figueiro, M.G. and Rea, M.S., 2011. Measuring and predicting eyelid spectral transmittance. Journal of biomedical optics, 16(6), p.067011.