My wife and I recently Netflixed a TV show called Daybreak. It's sort of like if Groundhog Day were a serious Detective show. The lead character is living the same day over and over again, and it's a heck of a day, with murders and conspiracy aplenty.
The show could inspire a really awesome RPG campaign, especially for a single-player campaign. You'd set up two or three mysteries, all happening on the same day, all with serious consequences for the PC if they were left unattended. In the background, you probably set up an additional mystery of a more sci-fi variety, that explains why the character is stuck in time.
The PC experiences each day over and over again, with things happening on a particular course if they do nothing. They need to solve all the mysteries, and then figure out how to resolve them all positively on the same day that they undo the time loop. You'd have a campaign with a definite end-goal in sight from the start.
There'd be a big burden on the GM, and I'd say you'd want to run it with a laptop at the ready, so you could have instant access to a lengthy timeline of where every NPC is at every minute of the day.
I'd run GUMSHOE for it. Both because I really like Gumshoe, but also because it has really easy mechanics to repeat actions again and again. When something comes up that goes poorly, you can spend a point to make it work out better next time. If you get a version you're happy with, a simple point spend ensures repeat performance when you try it again the next day. Very little dice karma to complicate things.
I'd have the PCs point pools refresh at the start of every day, so they had a limited supply of points to juggle for overcoming the obstacles. Experience would be tricky - as the cyclical nature suggests you'd never get to level up. However, if you watch the show, he does find ways to change things over time, and that might result in bonus dedicated pool points in GUMSHOE.
And I'd definitely use Daybreak's precedent for damage. Your whole body is doing the time loop, so if you're shot, you wake up with a bullet hole the next morning. Health and Stability would NOT refresh daily, so getting medical and psychiatric care may prove important if things go badly early on. That helps keeps the tension, and prevents certain Bill Murray style antics.
A tiny bit more about the show: Daybreak ran for less than a season. One season's worth of episodes were made though, and the last episode was clearly crafted knowing that they weren't gonna get picked up for a season two. It wraps up the plot neat-and-tidy, including several things that would have made a killer second season. It's really sad it got canceled, 'cause it's 5-star material, IMHO. Great acting, intelligent scripts, and plenty of unexpected twists. It's 58% mystery / detective, 40% action / adventure, with 1% humor and 1% sci-fi. If you've got netflix, it's well worth the rental.