What specs should I be looking for in a camera to record board-game sessions?
There are two approaches you can take based on budget.  The first is to record audio in camera, the other is to record audio to a dedicated audio recorder and mix it with the video after.  The former is cheaper, but the latter will generally produce better quality.

As far as features, beyond HD, you are mostly just looking for good performance in whatever lighting conditions you will have and then looking for good color and video quality.  Best bet is to look at sample videos from them to find one you like.  I'd also recommend taking a look at the GoPro, it's one of the better, cheaper video cameras available and would probably work well for your needs.

If you want something more general purpose, you could also look at a DSLR with video capabilities such as a Canon T4i which would allow for high quality stills as well as high quality video.

Your best bet for narrowing down your choices is really going to be to think about what else you might want to do with it and how much you would like to spend, then look at what is in your price range.

If you do want a dedicated audio recorder, the Zoom H4n is a reasonably cheap and decent option that will allow external microphones to be used for recording audio.