Of course, anyone would have to agree with ocnbrze, but now here we are.
Anyway, a green residue is a strong indication of copper. My best guess at this point is that the green oxide is appearing on some parts with a high-copper substrate that might be reacting with some nearby corrosive element, possibly leaking from a deteriorating battery. If the corrosion is extreme then the watch may have been sitting for some time in an environment that may have accelerated it. On other electronic equipment, old capacitors on a motherboard can leak a caustic substance that might cause damage to other components.
So you're concerned there may be some of that green crud that could have sloughed off into the works? Copper salts are often conductive. It might be moist and pasty or dry and crystalline, but if you can gently remove it, you minimize the chance of damage. I've salvaged all kinds of electrical and electronic stuff that had gotten wet. You want to get it open quickly, flush it out quickly if it's been in saltwater or something corrosive or nasty, spray wd40 on anything that looks logical for it, and dry it as quickly as possible. You've already got this thing open in a dry, controlled place, so just dry brush as much as you can see, then swab it finally clean with alcohol of at least 70 percent. I've also submerged boards in alcohol when more extreme measures are needed.
Once you're sure it's absolutely dry, put a good battery in. If it works, use it. If not, save the battery and put the rest in the proper disposal stream.
