What are you using it for? Do you want a pump, semi, single, o/u, sxs?
I like my Browning Gold ultralight in 20 ga. It uses an aluminum mag tube and is very light. I can carry it around following a quail sniffing dog all day.
H&R shotguns are pretty light. Single shot, though.
What sort of action type are you looking for?
The magazine tube material depends on the design of the gun, and the manufacturers may prefer the durability to weight and bulk in such matters, not to mention the additional costs imposed by new tooling.
The Benelli Montefeltro 20 is about as light a gun as I have seen that I would buy for a primary upland gun. They run 5 1/2 pounds or so. Recoil is pretty brisk for a 20, but they pack very nicely.
Usually doubles are going to be heavier due to their two barrels. The Browning gold that I mentioned earlier uses an alloy mag tube that cuts a half pound off it's weight. The Benelli at 5 1/2 pounds is pretty impressive but I imagine that recoil is pretty stout. The gold weighs in at 6lbs 6oz.
If you are not concerned about needing to have choke tubes, you may want to look a an older Beretta sliver or gold snipe o/u. I think the receivers were aluminum. They are very light (6 1/2 pounds about). There is one in gunsamerica for 599 currently. The Beretta Onyx series are also very light.
Just go with the NEF Pardner or H&R Topper and be done with it. Single barrel, and if you get a rifle, you can get extra barrels in about 30 different calibers including 12G. Or just get it in 12G.
www.hr1871.com <--- check them out. Walmart sells the NEF 12G for $98.
870's are heavy no matter what. A Mossberg 500C in 20 is a full pound if not more lighter than an 870. The 500C is the lightest 500. I have one for sale if anyone wants it, I just don't need all the 20's I have.