Ok, so it's coming on because the pressure is too low....another reason I hate dummy lights....and much prefer to have as many gauges as possible....but thats another topic.
My suggestion to you, change the oil and filter. What brand filter is on there now? If it's a Fram, that more than likely is your problem, as they've been known for being cheap and not filtering properly. I believe the internals break apart in the Frams, and therefore blocks everything up, causing low pressure.
Do you notice any unusual noises, such as knocking or tapping when you're sitting idle and the light comes on? If it's low enough to trip the light, you would think it would be making some sort of knocking or tapping noise.
As far as oil goes, i'd actually run 10W30 and see what happens. I run 10W30 Mobil 1 Synthetic with a Napa Gold filter (which is a Wix brand, one of the best out there and the Napa Gold only costs about $6 and some change in my area) and have zero issues. However, until you can narrow down your problem to either bad oil, a faulty sensor, or worse a bad oil pump, I wouldn't spend the $$$ on the expensive oil. I would get some regular house brand (such as Napa) 10W30 regular oil. I know the books, and most manufacturer's recommend 5W30, and IMO that's fine in the winter when the temps are colder. But now that it's getting warmer out, I think 5W30 might be just a bit too thin for the heat, especially once the engine gets warmed up. With that said however, I run 10W30 year round, and mine is usually garage kept, but has spent quite a few nights this past winter outside in single digits and teen's with no problems.
Where are you from, and how many miles do you have on your KJ?
I'd have to look up the oil pressure specs in the factory service manual, but I believe that they are pretty low for the 3.7, so if you're light is coming on like it is, you've got to have some seriously low pressure.
