https://www.youtube.com/watch?v=sVxqLBp6l1c
and
https://www.youtube.com/watch?v=CrD6hALcbkc

Check him out and watch all of his videos, it's worth it.

Post pics of the board.

I find that, unless you are doing it for fun, it’s not worth doing it at all.
If the corrosion is in an advanced state, the repair process will be very time consuming, and you’ll most likely have to fix some (if not many) damaged traces along the way.
Furthermore, unless you are extremely knowledgeable, chances are the board still won’t post, even after all the wasted time.
And even if the board does post... things will probably look very ugly with all the jumper wires.

If the motherboard is so bad, then you really will have to desolder parts to clean it properly - if you leave any corrosion hidden under an IC or a PCI or ISA slot, then chances are it will still continue eating the board.

Minor surface corrosion can be cleaned with vinegar and a toothbrush and scrubbing.

Heavier corrosion may need the bulk of the affected traces or parts cleaning with a fibreglass pencil - but these will quickly rub through many fine traces.

You can potentially repair minor damage by performing the cleaning and then re-tinning the area with solder. Bigger traces may need bridge wires to close completely removed sections.

Minor damage to ISA slots can be cleaned by scrubbing with vinegar and toothbrush - but if it has reached all the way inside, rather than just the tips, you're unlikely to get it all.

Oh, and of course, after cleaning with vinegar, make sure to thoroughly wash the areas with IPA or similar... otherwise you risk any traces of acid from the vinegar causing the same damage as the alkali from the battery!

I think, unless it's very, very minor corrosion, you need to be prepared to remove and replace multiple parts.

All of the advice here is awesome, I'm trying to get up the courage to fix an Amiga expansion board.

Here's another video that might be useful, for extreme cases. This guy fixes arcade boards that are heavily corroded. He's brave -- if not just crazy, he uses PCB Etchant instead of vinegar. O.O
https://www.youtube.com/watch?v=f439OuFz-eQ