I have an oily eyelid as well. I use Wardah pen eyeliner. Its a brand from Indonesia, very cheap, only about RM10 i bought it, its waterproof, certified safe cosmetics from Indonesian gvt and hallal.
To make it even more waterproof, I add Urban decay setting spray

just spray on your face after all makeup finish and its done.
No joke, even after hot weather, my face dont become oily..my face is actually very oily one..after few minutes washing face will get so oily