I couldn’t sleep and was thinking if there was a skill this doesn’t fit into the ordinary 5e skill list. There are lots of skills which don’t appear on the character sheet, but most of them fall into one that is already there. Juggling is probably performance, diving is athletics. I was thinking of skills which don’t appear.
Prescience is one. We know that this is a skill because it’s something that wizards can learn, and it’s not something that sorcerers gain innately. There are many ways wizards learn their skills, and it doesn’t have to be through schooling. Even barbarians can pick up some magic, accidentally. Why couldn’t your Fighter have found a note book as a kid in a box of pornography, abandoned in a bush somewhere. Nestled in the marginalia of the sensual artwork were notes on how to hone the skill of fortune telling.
Your Wisdom (Prescience) check allows you to attempt to understand the future, not through deduction but through divination. Your method of prescience may differ from another’s; it could include understanding tea leaves, meditation for some time, or maybe even glimpses of what lies beyond, coming unbidden and uncontrollably.
Peering through time is not entirely understood, and even those most dedicated to the art have been known to be unreliable. Your Dungeon Master may tell you of a cryptic omen or exactly what a foe will do next. They may chose to let you see miles away from where you rest or only meters around you.
So now that skill exists, there’s a few things to think about. The first is proficiencies. Obviously, this isn’t mentioned in any of the normal routes of gaining that tick on your skill list, like backgrounds. Ideally, this might be something the players can learn over the course of the game (using downtime) or give it them during a session zero – maybe swapping one of their normal proficiencies.
The other issue to tackle is that not everyone has access to this ability. You could instead make this into a tool, like the brewing tool kit. You need the crystal ball (worth at least 200 gold pieces) before you can even make use of the skill, for instance.
I prefer to keep it like a normal skill though; everyone can try it but some people don’t have a very high modifier. In fact, instead of the base modifier being 0 (plus zero on your d20), the base modifier could be -10. At first level, proficiency in prescience gives you -8 to the dice roll. The likelihood of hitting the DC to get decent information is low, and probably relies on getting a natural twenty. This makes the skill less useful until later levels.
Rolling 5 or less reveals nothing of the future. Less than 10 gives incredibly vague feelings. More than that gives gradually clearer images. It’s still, of course, totally down how the DM wants to deal with it, and what information (useful or not) they want to give out.
Of course, you could keep the default modifier at 0, but have the DC be set in the 20s or high teens each time.