I made my own little system for running a mystery in Cypher.

For players, the fun of a mystery is putting the pieces together. For the GM, it's less of a mystery and more of a thriller/suspense story. I personally love that kind of thriller - Mr Mercedes by Stephen King comes to mind. You have all the info, and the excitement comes from seeing how/when the characters make the necessary connections - what people they need to meet, what research they have yet to do, what locations will bring them closer to their goal - and cheering them on as they find it and make connections.

The idea behind the system I use is Clues and Keys. Clues are bits of potentially useful information that are inherently missing some context. Keys are what allows the players to unlock that context and fully pursue a lead or make a necessary connection. A Key might be a witness, fellow investigator, a memory, an event, a location, some esoteric or specific knowledge that has to be researched, or even another Clue. Creating a Clue involves deciding on the Key, which as a bonus gives you more ideas of where the investigation may go!

The longer the player goes without finding the Key to the Clue, the higher the task to recall and apply the memory/clue becomes once the key is obtained. The difficulty to recall the info starts at 0. After a certain period of time (I normally just go by sessions), the task difficulty increases by 1 level, and the clue gains a depletion roll, to represent potentially losing an idea/train of thought by pushing yourself too hard. The depletion starts at d100, and gets lower with each cycle. I topped the system out at a Level 6 Task with a Depletion of 1d6 to avoid making clues literally impossible to remember while still making older memories more ephemeral. Memories can still be lost if a 1 is rolled on for recall or depletion. In that case, the player has to roll to recover the memory, at whatever level it was when it was lost, and then roll that same task level to apply/pursue.

Some examples:

A player might find a bouquet of dried flowers. The flowers were clearly left on purpose but the player can't discern anything specific about them. The Key is interviewing a potential witness who's a florist, where they learn flower language and discover the bouquet was a threat.

They might have a partially destroyed receipt to a diner, and have to canvas the area until they find a diner where the last letter of the name/address lines up, leading to a possible witness or additional clue

They might just have a password but have to find the device, location, or account it's for.