'Name a Star' - Several companies offer this 'service' - it's a scam, pure and simple. The IAU is the only organisation with naming rights to celestial bodies, and they don't name stars. The companies that sell star names basically pick an unnamed star in the sky, and write down its coordinates in a book alongside the name the victim chooses. In theory this collection of names and coordinates is protected under copyright. But individual stars aren't protected, so there's nothing to stop me picking my own name for any star that's been 'registered' and publishing my own list - maybe I should create the imeem star registry?
If you hear of anyone considering paying to name a star then be sure to let them know it's a waste of money.
The IAU doesn't name stars because they're too far beyond the reach of human influence, it's conceivable that some alien civilization with a stronger claim to the property might have their own name. As such it would be impolite for us to contradict them. On the other hand, objects withing the solar system do get named by the IAU, there's a load of policies in place for naming objects depending on classification http://en.wikipedia.org/wiki/Astronomical_naming_conventions
Comets and Asteroids are the only objects that are regularly named - Comets are named after the people who discover them, and this frequently means that numbers have to be used to designate which specific comet is being refered to - e.g. Shoemaker-Levy 9
Asteroids are far more prolific - there are hundreds of thousands of the buggers out there, asteroids start of with a provisional designation based on their discovery date e.g 1997 XF11. Then after being observed for long enough to be sure of it's orbit it'll get given a number. Finally, numbered asteroids are eligible for naming, usually nominated by the discoverer. This is the main place where still living people get honoured with celestial names - although military or political leaders generally have to have been dead for a century to be considered. Some of the names are rather whimsical - 13681 Monty Python, 26858 Misterrogers, 15092 Beegees, 15845 Bambi and one of my favourites - 13070 Seanconnery. I have quite a few associates in the astronomy world with asteroids named after them, I guess mine is waiting for my mythical thesis submission.