Unless you write in a very strange setting, people are going to keep track of the passage of time. It’s a fact of life that we take for granted, but there’s a lot of history that goes into our own modern timekeeping system. January through June are all derived from Roman and Greek mythology, July and August named for Roman emperors, and September through December they just gave up and numbered (though they numbers are based on a year starting in March). Hours were generally agreed by most cultures to be broken into 1/12ths of a day (and a corresponding 12 hours per night, once civilizations cared to measure nights). Why 12? It varied by region, but mostly derived from a base-12 number system being standard. Days of the week are as much of a mess as the months. Sunday and Monday are literally Sun Day and Moon Day. Tuesday through Friday are all named for Norse gods (Tyr, Odin, Thor, and Freya, respectively). For Saturn Day, we dipped into Roman for some reason.

What Does My World Need?

For very primitive cultures, you can dance around the issue. Day and night, and a count of how many, can get you by in place of most calendar functions. Beyond that, you’re going to need to figure out how your people judge the passage of time, and decide how they would name them. As you can see from Earth’s example, our modern system is a bit of a mess. (of course, Earth’s worldbuilding is a mess). Here are a few things to consider.

How Earth-like is Your World?

Months are derived from moon cycles around here, approximate though they may be. What if your world has two moons? Or none? What if your moon is tidally locked and unchanging in the sky? Is your world’s axis tilted? That’s what causes seasons on Earth, and gives us solstices and equinoxes. You could have a greater or lesser tilt, or have none at all.
All these will affect how your people judge time. While systems will eventually become sophisticated over time, early on in a culture’s development, you’re going to have to get by with celestial observation. What seers and philosophers see in the sky can well set your world’s view of the passage of time.

How Do They Keep Track of Time?

The mathematical and technological means of tracking time will play a part in how your deal with terminology and idiomatic speech. If they don’t use clocks, then they won’t refer to time as “ticking.” Nor will time “wind down” if no one winds timepieces. If a society persists in the use of hourglasses, you might end up with phrases like “sand’s getting deep over here; will you hurry up?” or “who keeps flipping the glass? This is taking all day.”

Keep the technology in line with the culture. Smiths and machine makers are likely to build clocks like we knew them centuries ago. But a culture that reveres stonework might use sundials into the modern era, with sufficient modification. Maybe that culture never becomes obsessed with exactitude in measuring time and becomes widely regarded as a patient, relaxed people.

When you decide on the mathematics of time, you’re on your own. Keeping the 24(12)/60/60 timekeeping that we use on Earth is no sin. Your readers will likely gloss right over it and silently thank you for taking it easy on their brains. If you’re looking for a challenge, feel free to tinker with different number systems, more or fewer distinctive units of measure, or days that are different lengths. Any numeric value in your timekeeping is fair game to change. Just be sure that you don’t complicate things to the point where you can’t keep track of it any longer.

Naming It All

When you have your system settled, think about who or what your civilization is likely to name these months, days, or years after, depending what you want to call them. Also consider who gets to do the naming (king/emperor, scientist, international convention, etc.).

Some ideas:

• Scientists or philosophers who were involved in working out the system
• Deities
• Monarchs
• Natural phenomena
• Bland, prosaic descriptions that took on their own meaning after changes in language (see: September through December)

In the end

Remember, as with all worldbuilding, the important thing to remember is that you’re trying to impart realism to the world in the mind of your readers. You’re not out to actually create the world, nor are you trying to win a cleverness contest. Consider the following as a caution:

• Earth: “Today is February 25th, a Tuesday.”
• Fantasy: “It was ninth-hour on the fourteenth day of Melukan. It was also Sebbu again, and every Sebbu we all gathered at the Temple of Meluk for prayers.”
• Cleverness contest entry: “At twelve parsepts to the  jahn, on the halfday Rezaa of the Il-Talan, I checked my gullau. In another eight or nine dekendi, it would be Chahl.”

Cut your reader a bit of slack, and give them some ideas of analogues to the terms you use. You can gather from the middle entry that Melukan is probably a month or a season, named after the god(dess) Meluk. Sebbu sounds like a day of the week. In the last example, I’m not even sure what’s going on, and I wrote it. I think a gullau might be a wristwatch (or maybe a microwave, or an alarm clock).