Oooooo, this is a great topic idea! I used to be a mental health evaluator in the criminal justice system, so true crime elements in all media forms have a significant appeal to me. So many of the cases in LA Noire come to mind, the Phoenix Wright series, Batman: Arkham All the Things' use of detective mode and clue finding, those Sherlock Holmes adventure games (demon Watson notwithstanding), even the crime scene investigations from Heavy Rain...
Most memorable for me though is probably from Star Wars KOTOR - the Jedi investigation into a murder on the Selkath homeworld of Manaan. After landing on the planet, you get sucked into gathering evidence and weighing in on the legal proceedings of a Republic sympathizer who is accused of murdering a woman with known Sith connectioins in a dingy (as dingy as that era of graphics could allow) hotel room. Was it political manuevering, an assassination, a spat between lovers turned violent, or a freak accident? You get to collect evidence, interrogate witnesses, and use the Force however you see fit (okay, lots of mind manipulaiton) in the process, and during the trial, you can even withhold or falsify evidence to justify your ethical allegiance(s). I specifically remember this one because when I played through KOTOR for the first time, I was still in college waaaaaay back in 2003, trapped at home with my roommates during an ice storm. We played through the entire game over the course of a few days, and this came right after the big twist involving ********** - forced me to re-evaluate who is this character going to CHOOSE to be in the process of investigating the crime with the knowledge he now has of his identity. Good stuff!!!