There have always been shades of Groundhog Day in video games. The save/load mechanic is functionally identical (with slightly more user control and less Sonny & Cher) to Bill Murray waking up each day afresh, without consequences from the previous day. Edge of Tomorrow explored that connection really well, but I'm not aware of any games that have been inspired the other way around.
Until now, says the voice of all your favourite infomercials. Garbage Day is on Steam Greenlight right now, and its central time-loop mechanic is consciously inspired by Groundhog Day.* You wake up the same morning after morning, no matter what you did the previous in-game day. What can you do? It's an open-world sandbox game, and the developers seem committed to providing as many options as possible. You can "walk or drive to work; go take a shower; eat some cereal; watch your VHS tapes on your VCR, chat with your neighbours, [or] murder innocent people, because you're stuck in a time loop so don't worry, they'll still be alive in the morning." So pretty much everything you'd do ordinarily.
The murder and violence options I can take or leave, despite always having wondered whether Bill Murray indulged in any of that. I can get that stuff in other sandbox games if I want it, and there seems to be a satisfying range of other things you can do. Hey, I find performing menial tasks in video games bizarrely enjoyable. But what really interests me is the potential implications of the time-loop mechanic. The goal seems to be to escape from it, but how? How do multiple "playthroughs" interact with each other? How long can I go on without exhausting gameplay options? Can I rescue kids falling out of trees? These are important questions. I'm genuinely intrigued at what Garbage Day could be. It remains to be seen what it actually is.
* One has to hope that the title is inspired by the greatest scene from Silent Night, Deadly Night 2: