What sort of time/date system would suit a Dyson Ring, also known as a Ring-World, an artificial habitat megastructure shaped like a circular band, located in the habitable zone encircling a star, in this case Sol?
What about a Dyson Sphere?
In either case, what factors would decide where to put the Prime Meridian?
The answer is more social than structural, once you reach a certain point. When you're on a rotating planet, obviously there is a cyclical time system which we cannot change very much: the rotation of the planet. However, once you free yourself from such constraints, doors open.
The most likely answer in many science fiction books is that you will assume the time system of the planet of origin for the species that builds the ring. Most likely the species will have evolved some level of interconnection with the natural cycles of their planet and maintaining that cycle will be beneficial for health and well being. It is unlikely, however, that they would include timezones, which would require a prime meridian. If you really get down to brass tacks, timezones are pesky things, holdouts from the railroad era forming a compromise between wanting "12 noon" to mean the sun is overhead and the desire to easily be able to tell when the trains were going to arrive. Without all that pesky planetary rotation mumbo jumbo to get in your way, there's little to no advantage to having timezones. It would be the equivalent of everybody operating on UTC time.
Speaking of which, they might not want a cyclical time system at all. There's a lot of advantages to them, but there's also lots of limitations. For highly technical work, it's common to measure time in seconds since some epoch. For example, most of our high precision timing is built around an "epoch" at 1 January 1977 00:00:32.184, also known as Julian Date 2443144.5003725. TAI, UTC, and several ungodly other ones like TCB all were synchronized at that moment and we've worked from there ever since (the absurd fraction of a minute bit was done to preserve continuity with Ephemeris time... metrology of time is fascinating but ugly at times)
All of our modern computers use this approach as well. Computers don't need a cyclical time system, so they just count straight.
So this should point you to your first hint as to what your dyson ring might use. How important are computers to keeping the civilization running? The more important they are, the more likely they will use computer friendly free-running time, rather than some complicated cyclical system. The more important the originating species is, the more likely you will see carryovers from their home planet.
Niven's original ringworld had a smaller ring of shades circling the sun which created a day- and night cycle.
With no such artificial mechanism, there is no day- and night cycle. A ringworld usually rotates to simulate gravity. The sun will likely be so bright that individual stars will be hard to see, but some very bright stars in the direction of the rotation axis might still be visible. They would serve as an indicator when the world made one rotation. How long that would be depends on the radius and the desired simulated gravity of the ringworld.
In total absence of any cosmic reference system, the answer to the question is the same as "How would people tell time if it was always day?".
A culture which is scientifically advanced enough to realize that they are living on a ringworld and that the reason they feel gravity is centrifugal force will likely be able to calculate the diameter and rotation frequency of their world even without having to develop astronomy. But it is questionable if their system will catch on when there already is a more established traditional timekeeping system.
Placing the prime meridian would be just as arbitrary as it is on Earth. The only significance of the Prime Meridian on Earth is that it goes straight through the building in Greenwich where the coordinate system was invented. A navigational system on a ringworld would likely also have a reference-point picked for its historical or political significance.
Like Philip said, Niven's Ringworld had an inner ring of shades that rotated inside to give a day night cycle.
The inner ring was not a solid ring. It was made of electricity generating sections connected by unbreakable monofiliment lines.
The downside of this setup is that you'd still have to have time zones to deal with local time.
It might be interesting to instead have the inner ring be solid, and oscillate up and down in its orbit, so the entire ring would be on the same time.
If you're going to go to all the trouble of building an impossibly large structure out of impossibly strong materials why give yourself the same drawbacks as a rotating sphere.
