Why is resistance increasing while I'm measuring using a multimeter?
After you run this test for 10 minutes, wipe off the water and take a close look at the copper where the water drop was.  You will see some discoleration.  Essentially the copper was corroded a bit where the water drop was.  As you probably notice, bare copper left in the elements is no longer that bright copper color after a while.  The same thing happened to your copper electrodes, except that the electric current speeded up the process.

The reason the resistance goes up is because the corrosion layer has significantly more resistivity than copper.  This is one reason the mating surfaces of electrical connectors aren't made from copper.  They are usually made from material that doesn't oxidize, like gold or nickel, or forms a conductive oxide, like tin.