Ok, I've delayed updating this thread because I don't know what went right. Seriously.
First, what I know - I have twice successfully electroplated what appears to be copper onto what I believe to be tungsten wire, and I know that after I plated it I was able to solder it easily. Before that it would just reject solder.
I dissolved copper sulfate in water the first time; the second time around I did the same but eventually added some dilute acid. Both times I used a couple of nearly-dead AAA batteries to provide a weak ~1 volt current between a copper wire cathode and a tungsten anode.
But here's the problem. On both occasions, I met with no success for a long while at the beginning. The tungsten was clean; on the second attempt I scrubbed it with alcohol and scraped it down as well. But I had it plugged in for hours with no visible copper deposition. Someone half-jokingly suggested that I replicate the steps which had led to my initial success; I had once tried connecting a 9-volt battery and was quickly met with a rapidly growing black deposit on the tungsten, as well as a great deal of bubbles from the electrolysis of water. With my best scientific reasoning I figured "why the hell not" and hooked up a 9-volt. Of course there were bubbles, of course the black deposits appeared. After a minute I wiped the tungsten clean and reconnected the weak AAA batteries, exactly as before.
Within seconds, I had a layer of copper.
What exactly is going on? Did I blast something clear of the metal surface? Did I inadvertently reverse some corrosion, or force copper ions into the solution? Why in the world would a brief pulse of relatively high voltage make a difference?
For what it's worth, I have pictures. The first attempt was in a glass beaker, the second in a Tupperware, but I think I will go back to glass. I'll post the pics in a separate reply to keep things neat.