Step 1: Buy new oil pressure light switch, they're cheap. Make sure the wire isn't grounding out on something.
Step 2: Clean oil pressure relief valve.
If this fixes it, then the valve was sticking. Sometimes the valve hangs up because of wear, but usually it's a bit of crud. BUT, please clarify the original description, is the light OFF or ON steady as you increase RPM?!? That is, it is on most of the time, but almost goes out at idle. That would point to a supply problem. Is the engine a sludge bomb? You could have a bunch of crud sucked up against the pickup screen, in which case then...
Step 3: Remove valve cover and clean everything. Is it clean under there or nasty? Does it look like oil is getting up there? You'll want to remove the rockers and shaft. A wire brush and shop vac will work OK. Oil everything and adjust the valves.
Step 4: Put in a whole can of SeaFoam and cross your fingers. You may get lucky and the sludge will soften and pass through. If so, keep an eye on the oil, and change it when it starts looking dirty. If not, then...
Step 5: The pan comes off. Kind of a PITA. If the engine has been run very long with low oil pressure, the bearings are probably pretty beat. Then again, if I was pulling the pan, I'd just plan on replacing the bearing inserts and oil pump at 160K.
As for oil, thicker is seldom the answer. At best it will mask the real problem. I run Shell Rotella T6 5w40 (synthetic diesel oil) in just about everything I own. If you are flushing the engine, or it leaks/uses oil, then wal mart house brand 15w40 diesel oil would be my pick.