Earth-like planet orbitting red dwarf.
Survivors from the Earth.
(Only one planet, not post-planet civilization like in question: How Would a Post-Planetary Civilization Measure Time? )
All typical units like day, (lunar) month or year have no longer much meaning. Needless to say local year does not have any round lenght.
If one tried to redefine second, then would face redefinging all SI units and cause serious rounding errors. If one tried to redefine year, then all historical data would start being problematic.
No longer day-night cycles, presumably the same infrastructure can be used in more than one shift.
Our contemporary way of measuring time is not specially logical:: 365, 12, 7, 24, 60... etc. It would be nice doing something with it by occasion.
People tend to react poorly if their day-night cycle departs seriously from 24h (actually studies show that average day would be a few minutes longer if one asked our biological clocks)
(the only simple thing is what to do with computers... well there would be no reason to drop unix time)
Issues to deal with:
1) Design calendar (what to keep? year? week? Round something a bit???)
2) Design a day cycle (lenght, how many shifts? 2 to allow some margin of error? 3 to perfectly utilize all equipment?)