DON'T USE VINEGAR. [I know I posted about doing just that before - but I know better now.]
Do as follows:
1. Try to clean the spot as much as you can with a dry cloth and then some rubbing alcohol.
2. Mix baking soda with some water to a paste and apply it to the damaged parts with a toothbrush.
3. Let it sit for a short while (20-30 Min.).
4. Clean with distilled/demineralized water.
5. Dry the board. (Hand-dry with a clean and soft cotton cloth. Easiest way to make sure the board is really really dry is heating your oven to ca. 70ーC, then turn it off (!) and put the PCB inside over night.)
Here's why: Mixing an acid (vinegar) with a base (leakage) could lead to a chemical reaction that will further damage your board. Also the proportions of battery leak to vinegar would have to be perfect to really neutralize the leakage and not leave some of the vinegar on the board that will eat through the metal even faster. Backing soda (sodium bicarbonate) is amphoteric, so it will neutralize bases and acids.
Thanks to the people schooling me on this. Read further
here and
here.