We have a slate floor in our sunroom (which is the gathering place when our children and grandchildren are visiting.) I found that regular cleaning did not remove oily looking stains caused by food being dropped.
I decided to try a little trick I have always used to get oily stains out of clothing: sprinkle baby powder on the stain, rub it in, and let it set for a while - sometimes it only takes a few minutes, but on tough stains it takes longer, maybe even several hours. I have also found that cornstarch or flour will sometimes work if I don't have baby powder.
After letting the powder set, brush it off or vacuum with the brush attachment, then wipe with a damp cloth. It's worked many times for me! (On clothing, it usually is not necessary to wipe with a damp cloth.)