We get asked many times a day, "Do I use grease or oil when lubing the hinges of my new over under?
Short answer: Grease
When lubing the hinge points of our over unders or side by sides we like to use grease, not oil. Oil, as it heats up, becomes thinner and could run out of the areas where it is needed most.
Lubriplate or white lithium grease is our preferred, but really most high temp greases will work. Grease tends to stick around longer than oils do, keeping your hinges from galling as well as making it easier to open and close your gun. You want to make sure you put a good smooth coating on all of the metal against metal bearing surfaces of the firearm. Most of these surfaces are where the barrels fit into the receiver of the firearm. In the illustrations below, you can see Beretta 68X series barrels and receiver with and without grease. You won't need quite that much grease when lubricating your own gun, we used extra for visual effect.
Picture 1 above - As you can see, there is no grease on the barrel yet.
Picture 2 above - Using plenty of extra grease for illustrative purposes, you can see you want to put a good coating on the sides of the barrels as well as inside the hinge point.
Picture 3 above - The left hinge of the receiver is shown without grease, the right side is shown with grease. This area is very important to keep lubricated as the pressures here can create galling very quickly without any lubrication.
Each brand is a little different as to where it needs to be lubricated and the manuals will generally give you a good idea of where you need it. A little extra is probably better than not enough, so when in doubt, a little more likely won't hurt.
Almost as important as putting grease on the gun is cleaning off the dirty grease when you are done shooting and re-lubricating with fresh grease. Not changing the grease can cause as much damage to your gun as not greasing it at all. Dirty or old grease can contain grit and contaminates which can scar the action. Over time and with use the properties that make grease work will degrade and become much less effective. So be sure to clean and re-grease.