Games that help me cope with pain are similar, but require a more specific type of game to be effective. SnowWorld, the VR game that helps burn victims cope with the pain of skin treatments, focuses on total immersion. The patients wear headsets and noise-cancelling headphones. When compared to a game like Mario Kart, SnowWorld is much better at distracting from pain. I know that games can be a powerful distraction, and I think a game's ability to distract depends on the genre of game and the player's experience with that game.
For me, the best pain-distraction game is Lumines. It's a falling-block game where the difficulty waxes and wanes with the soundtrack, and I'm proficient enough to play for 30-40 minutes before losing. I loaned out my Vita and struggled to find a good replacement game.
Tetris and Dr. Mario got too difficult too quickly for me to "zone out" to them. It made me wish that either game had a fixed-difficulty "endless" mode. I also tried out a match-3 game Zoo Keeper, which had many of the qualities I sought: it requires continual attention, the task is not too difficult and game sessions last a long time. However, as you level up, you have less time to find a match; it made me panic (it's otherwise a fun and cute match-3 game).
I tried out some other types of games to help with my pain. The DenpaMen,* a dungeon-crawler where you sit around watching your Denpa men automatically fight for you, was pretty good to zone out to, but it doesn't require enough of my attention to make me forget I'm in pain. RPGs are pretty good, but when the story gets going it's like I'm reading a book, which isn't as... interactive? So I've been trying to play more puzzle games in hopes of finding a good "zone out" game. Any recommendations?
*listen to parts of The DenpaMen soundtrack here... it's cute.