Supposing I have an unknown lens of which I don't know f/# and focal length. I want to estimate the total amount of light that my lens can collect which, I believe, is given by the solid angle that my aperture subtends with the object plane. Is it correct to measure the size of the aperture by imaging (using a camera with a lens) the size of the entrance pupil of my unknown lens? That could be done by placing a ruler next to the unknown lens and moving it to be at the same focal plane as the entrance pupil. From the size of the entrance pupil, can I then calculate the solid angle with the usual maths?
If you don't know the focal length or the aperture, there is no way to figure it out. The amount of light that a lens is going to let in depends on the field of view of the lens as well as the size of the entrance. You don't know enough for the field of view, so you can't know how much light it gathers.
An alternative method that would work would be to make an area light source of a known size and place it a known distance from the lens. You could then measure the size of the circle project by the lens. This would allow you to determine the focal length and field of view, then you can perform the rest of your calculation to determine the speed of the lens.
Trial: Nikkor 50mm, f/1.8
Measured diameter of optical path looking into lens ~= 28mm
f reference plane
1.78 lens to body seating plane 2.5 front element surface 1.5 rear element surface.
Trying this with a number of lenses of known aperture may give a guide as to how accurate this method is and which measurement planes and diameters should be used. And may not :-).
A 50mm lens with a 50mm effective aperture has a f-number of 1.0
If your lens is a 100mm lens, if it's front element (the entrance pupil, effective aperture of the lens) measures 35mm then its maximum aperture is f/2.8
In theory the aperture blades, which are not perfectly rounded, will slightly reduce the amount of light that passes through the lens, introducing a tiny error in the calculation, I am not a mathematician but I guess the error is microscopic enough to be ignored.