|MadSci Network: Astronomy|
Luke, great question!
You're asking about the fascinating scientific field of Astrometry ("star-measuring").
There may not quite be as many reasons to measure stellar positions as there are stars in the sky, but there's still quite a few of them. The most obvious and oldest is measuring the time of day. Early clocks were so inaccurate that they could be reasonably set by comparing them to the Sun--but once people built clocks that could tell time to the second, have you ever wondered how they knew what time to set them to?
The answer is by the use of an instrument called a transit, which is basically a telescope carefully mounted to point only along the meridian at its location. By looking through the telescope, an astronomer could note the passage of a star across a thin vertical line and note the exact instant of passage. Then, by knowing the position of the star very accurately, the moment of passage can be converted to a time of day for the location of the transit. But, as you may have noticed, the accuracy is time is dependent on how accurately the position of the star is known. This includes know the positions of the stars relative to each other as well as the absolute positions of the stars in the coordinate system of the sky.
More recently, measuring the positions of the stars accurately from year to year not only allows us to find out their Proper Motion but also allows us to detect possible planets in orbit around them by noting the slight "wobble" in the star's paths.
Knowing a star's position is three dimensions, instead of just in the sky, allows us to learn about the motion of our portion of the Milky Way galaxy, and learn how stars like our Sun, and others, are formed.
In 1989 a spacecraft named Hipparchos was launched whose sole purpose was to accurately measure star positions above the distortion of Earth's atmosphere. Look at all the different subjects of the scientific papers written using the data from the mission.
Last but not least, the Hubble Space Telescope could not be accurately pointed without its catalog of 18,000,000 guide stars, which it uses as a reference to keep it aimed anywhere in the sky it's pointed. The Guide Star Catalog is one of the most comprehensive and accurate list of star positions compiled to date.
I hope these answers give you some ideas for further research in astronomy.
Try the links in the MadSci Library for more information on Astronomy.