We'll still use the second as a frequency base, defined in terms of physical processes.
To note time in a portable way, like making sure your ship reaches the planet in its orbit, the time in seconds as used by Unix. What reference frame is the time noted in? For purposes of logistics and commerce in the solar system, a suitable reference frame would be devised. It could be much simpler than what is used on Earth now (for GPS etc.) because Earth is rotating. A simple base would be to treat the the sun’s position in space, as measuered by a hypothetical clock that is not rotating or moving with respect to the sun.
But the center of the sun is not really a natural place to measure from. Lets model the sun as a hollow sphere with the mass in the shell and subtract out contributions from the gravity of the planets, and imagine a clock motionless anywhere inside the hollow part.
But that’s getting complicated and the model is limited by observational accuracy and changes over time. Scratch that.
How about looking at distant pulsars. Model their cadence factoring in the orbital mechanics of the system and its rate of slowing. Have several so if one changes due to an anomaly it can be removed and recalibrated. Combine these adjuated frequency bases in a manner similar to what is done with atomic clocks now. Define the time not for the center but for a theoretical distant observer, and that value is adjusted based on your position in the solar system (general relativity) and motion (special relatativity).
Still, the accuracy possible will never be as good as local clocks measuring local time, and technology will drive improvements based on what is possible.
Plotting events in spacetime for data recording will use an single number, counting seconds. But what will humans use? All kinds of things. Mars uses local sols because solar power is tied to the natural rotation. People living somewhere may have cycles that are meaningful to them and those pervade all activity. People generally live in a 24 hour cycle due to biology, regardless of local conditions, but that might be a matter of choice if some simple pill or environmental cues can make normal people cope with (say) the Martian dinural cycle.
Cycles similar to “weeks” appear to be useful, as civilizations have had 5 to 10 day periods for work schedules and personal planning. So whatever can be made to fit with the imposed primary cycles, somewhere in that range, will be a week and the term is deliberately non-standardized and used with local convention.
There is also a desire to sync up with those you’re working with, which is why bases in Antarctica use the time zone of their main supply chain and ignore the correct longitudinal zone. Do astronomers at the south pole (e.g. BICEP experiment), expecially the over-winter caretakers, follow a 24-hour schedule when the sun never rises and the machinery runs non-stop? I’ve never asked.
Computers will have no trouble stamping your log entry with a universal “star date”, and for scientific measurememts it might be needed to indicate your world-line time rate, but that is not needed for human scale events.
If week cycles vary, local custom might name the days creatively with intention to have personal flourishes. Or, a standard may exist for names for days in weeks of each length, or some combination of starting with the common names for work-week but naming days of local significance specially.
To summarize, a colony or worksite may have time cycles of significance, which may be planet rotation period or something completely different. Human timekeeping built around that will be varied, local, and a huge mess. Official timekeeping will be a single scalar number.