Hi Mark,
Go through [guide|12457|this teardown], pull a picture showing the broken connector, and post it here.
If however, it is the mini-USB charging port like you described, you'll need to replace it using solid soldering skills, the right tools, and of course the right replacement part. No need to replace the whole board. My advice in this case, if you lack the pre-requisites, is to have a specialized tech handle the repair, or take it to a GoPro reseller and have a shot at warranty coverage.