Okay, so humans are finally transitioning to a space-colonial society. We sent ships out, colonize worlds, etc... We aren't really in the business of sending multigenerational ships. Instead we use time dilation. We generally only go somewhere when we can get there within a year of subjective time (although probes can take longer.) The only potential problem would be that then the ships would far outlive the people on the planets, and all you get is a bunch of ships shooting everywhere, with no real centralized civilization or cultural. I call it potential, because it has long been solved: time dilate the people back home as well. Most bases our put into orbit around gravitationally strong bodies. Shooting a small black hole into a gas giant usually does the trick, although even some suns have been used this way if the inhabitants are rich enough to entirely forgo solar power.
Now we have a problem: keeping time. Note that it isn't so much a technical problem: computer get along just fine calculating each other's velocity. The problem is people adapting to the fact that time is different everywhere you go.
For an example of the confusion. Let's say you are going to visit gravity park in New New New York solar system. Your tell your friends you will be gone for about a week. Is that a week of the parks time, your friends time, or your own time?
For another even more complex example, consider that your space car breaks down, so you get it towed to the space mechanic. The space mechanic says it will take about 10 days to fix. 10 days for whom again? (Both you and space garage, and space mechanic can undergo different, and varying amounts of time dilation between now and then.)
How would how we describe time have to change in response to this?
- Most reference frames humans travel in are within 30x of each other, depending on wealth and occupation.
- An individual typically will only experience a 5x difference between fastest and slowest for their weekly commutes, and maybe up to 10x difference if they move (but stay in the same socioeconomic group and occupation.)
- That is to say, an observer at infinity would observe a rich person's clock typically tick 30x slower than a poor persons. A person clock tick between 1/10th to 10 times the rate after they move to a new home, a person clock tick 5x faster at the fastest part of their weekly commute to the slowest.
- Generally, ships and bases don't follow any regular rules. A computer can compute it no problem, but if you are taking a space bus that makes regular use of black hole gravity slingshots, time dilation throughout the journey can change in a way that is hard for humans to keep track of.
- Changes to language shouldn't to inconvenience people in their daily lives. It shouldn't be complicated to say "here, watch this 3 minute video," or "give me a minute to figure out the answer" if you are face to face with someone.
An answer should address both changes to language (both time unit words like "second" and "week", and also somewhat relative words like "then", "now", "before", "after", etc...) and the technology to make it easier to deal with (what the heck would should your wrist watch tell you?)
Note: Don't forget to take relativity of simultaneity in to account. That just messes with things more.