Ownership of my games: Either DRM-free, so I can copy them, or at the very least, future versions of the same console shouldn't make me repurchase the same downloadable games.
Don't require server connections for single-player games.
Ease of use: Click 'download' on a game I purchased, and I boot up a different game while the first game downloads in the background.
Doesn't try to trick me: Don't make me buy your own currency. If I want to buy a $7.50 game, don't make me buy $10 worth of your money.
Any game available on disk should also be available for digital purchase (and buying a disk gives the digital version as well). I don't buy games at $60 or $50. If the game never decreases in price, I'm either going to buy it used (unfortunately giving you no money as a developer) or not buy it at all.
[minor change of topic]
This is why I buy on Steam. The people who want to play the game the second it comes out, they feel it's worth $60 to them - that doesn't mean it's worth $60 to everyone. After a year, it should be $40. After another year, $20. From time to time, it should go on sale for $10-15. That's when I'll pick it up... Not because I don't want developers to profit, but simply because very very few games are ever worth $60 to me personally. When I do buy a game for $15, then I want the money to go to the developers. By not decreasing the prices a reasonable amount over time, it's the developers (well, okay, the publishers) who drive customers to used-game sales and (worst case) to piracy.
The annual amount I spend on games? Less than $100 total. If your products aren't within my price range, then I won't buy your games, or I'll buy them used. I'm fine buying 4-year old games - but if you don't make them available to me (because once they stop selling, then you stop producing disks) then I can't buy them. Digital distribution has reduced your: A) disk manufacturing costs, B) distribution costs, C) middlemen costs. Instead of $50 dollar games decreasing in prices, they rose to $60 prices instead. Okay, so I'm oversimplifying, and ignoring rising game development costs, rising marketing costs, and the economic downturn. But if the economic downturn hurt studios so badly, how do you think they hurt consumers? The response should've been cutting costs, not raising prices, otherwise you drive more customers to used-sales and piracy.
I don't want $1 games on the iPhone either. Give me a happy middle. $10-$20 games (several years after release) for 10-20 hours of gameplay. No hidden fees and no monthly fees.
Other than that, lots of cooperative splitscreen games. I like local coop and local multiplayer.