You may be interested in the 'Hours, minutes, and seconds' section of the Popular Questions pages (as found in the sidebar).

In Egypt, the Book of Gates uses stars to predict the time of sunrise (or more accurately, groups of stars). The problem is that stars are not constant - they don't rise at exactly the same time every night (there's a 4 minute successive lag every night) and you'd need exactly 12 stars at specific intervals. Every 15 days you'd have to start all over again (due to the 4 minute shift).

So the idea is that each guiding star (or bright star) has an attendant group of stars (to help identify when it's coming), and appears at a particular place or gate on the horizon. There are multiple versions of the Book of Gates, some which have different patterns, different stars, and different gates - so the night-time hours were not a constant fixed length.

Senenmut's tomb has a wonderful example of a star clock which shows the complicated nature of the calculations.

In the 15th century BCE, we have the earliest description of a water clock and hours were measured like that. A vase was filled with water, and had a needle-sized hole in it. As water dripped out, the inside of the vase revealed a scale which would tell you how many hours it had. Daylight used shadow clocks and probably merkhets which served a dual function. You were limited in the time you could use this of course, and a cenotaph from Sety I indicates that you could only used this the 4 hours either side of noon.

As for words, there are words for 'hour' and for segments smaller than an hour, but not for 'minute' or 'quarter hour' or 'half-hour'.