You could play any kind of turn based games over planets. In the future even non-turn based games are going to be played, although indirectly.

A fun game would a real time military strategy game where the delay for both planets is artificially made the same. Then the command delay becomes part of the game, but the game still runs in real time with information updating as soon as it is available.

People aren't going to stop playing chess or go, after all.

That reminds me of Neptune's Pride, which is a game that leverages latency for dramatic effect: http://np.ironhelmet.com

Of course, the dynamics of video-game playing in zero-gravity should be interesting. (Do optical drives work in zero-g?)

Yes, why wouldn't they work, they don't particularly rely on gravity to operate. But then the question is why use optical drives at all?

Because mechanical parts (and the way they are controlled) sometimes are built to compensate for gravity, so the absence of it might change the way it functions.

Optical drives work on their side or upside down, which suggests to me that they'd be fine.

But how long will they work compared to the right way up? I can imagine that running it on the side will have unequal distribution of forces and thus wear for example.

There are consumer products that have them mounted sideways (e.g. I think one of the xboxes?), so it seems like they're reliable enough for consumer electronics standards.