Somehow oil or something got splashed on the painted drywall and now there are two stains present. I was wondering what tips people have to attempt to remove the stain, or is the simple solution just a repaint job?
Salad/cooking oil of some kind.
If it is a lot of oil you may need more than paint. I would go over the area with something like drylok and let it sit for a week or two. If nothing comes through then just repaint over the drylok. If you see any sort of oil coming through - even a tiny bit - then I would skimcoat the area with drywall mud. Then repaint after that. I have personally tried to repaint over oil stains totally wasted my time, noticing the same oil coming through a few days later on new paint job. I am sure this has to do with what kind of oil, how much, and the paints you use (oil based paint will hid this better than latex).
If the paint is good quality latex or (any type of oil paint), try wiping the stain off with a dry paper towel or absorbent cloth. If that doesn't do it, add a little dish soap and water and scrub vigorously, then rinse and wipe dry. Water won't harm the wall or paint if it is only done occasionally.
If, after letting the area dry well and oil is still visible, the next level of escalation is TSP, available from paint and hardware stores, larger grocery stores, and others.
If that doesn't do it, priming and repainting is called for.
There ar e numerous grease cutting detergents, such as Fantastik, that are often very effective at removing both colored and clear stains from painted surfaces. While these can be sprayed on, I prefer to test by putting a little on a soft cloth and rubbing gently.
If more is needed, and if there is no sign of paint coming off on the cloth, you can try a direct spray and remove immediately.
Next level is to spray, allow to sit for a minute or two, then rub.
Although these cleaners claim to not need rinsing, I always do.
If that fails, you are probably in for a more serious scrub that may require a repaint. If that is the case, I would prime with a stain blocking primer. Like Kilz or you risk a bleed through of the stain.