Right now one of the most important features to me is the ability to suspend and resume a game whenever I need to. I have 2 little kids (under 4 years old), so when I'm at home they always take priority. When I game I need to be able to stop at any moment to help them out or spend time with them. It's given me a completely new appreciation for good suspend and save features.
Save anywhere is really nice to have, especially if it's quick to do. But even better than that is the ability to instantly suspend a game. This is why I've been spending so much time with my 3DS and Vita. With the 3DS I can just close the lid and know the game will be in the exact same spot when I come back. The Vita is just as easy with the PS button which instantly suspends the current game. This is easily the best feature of these systems for me.
Another awesome thing I've found is on the Wii U and 3DS virtual console. Virtual console titles are games from the NES era through the N64 era that are emulated on the current Nintendo systems. They have the idea of restore points. A restore point can be created with the push of two button (one button gets you into the virtual console menu, one creates the restore point). Because they're on virtual console it's possible for me to go back and enjoy some NES and SNES classics since they now essentially have a save anywhere feature. It's very similar to save states in most emulators on PC, but it's fantastic to see Nintendo support this idea on consoles!
The reverse side of this is playing games without a good save system or playing games online. If I want to play those types of games I need to wait until late at night when I'm positive my kids are sleeping. This limits my play time to around 2-3 hours at night and that's if I do absolutely nothing else with my night (which is rare). So, while I love Hearthstone and Heroes of the Storm right now, they're games that I can't play very often. There are limited times when I can play them. I can't even imagine trying to play an MMO at the moment. It's totally infeasible.
I don't begrudge my kids for a second. They're amazing and they'll always take priority. And they're turning into little gamers themselves. Pretty soon I'll have a little Player 2 and Player 3 all the time! But it's just amazing how much kids have changed my gaming patterns.