Can a star be named after me?
Naming of objects in the sky is subject to international treaty and (when the objects are close or bright) occasionally acrimonious debate. Names are set only by the International Astronomical Union, basically the United Nations of astronomy. Hence, when you receive an ad in the mail or see something in a magazine offering to name a star after you or a loved one for a fixed price, even if it contains the term "International" in the ad, you should react with skepticism. Although not illegal in most states, this practice is certainly unethical if not enough attention is paid to the fact that such naming is not official and conveys no rights.
Why is it unethical? If it doesn't hurt anyone, how does it violate ethics?
One of the basic laws of ethics is that before you sell something, you should own it. If all this practice did was generate joke gifts, it would indeed hurt no one. However, many people buy these star names thinking that they are making a lasting monument to a departed loved one or terminally sick child. Think of how low they feel when they are finally told that they bought nothing of value from a company that did not own what they were selling. No one will ever use such purchased star names. They are worthless.
When does the International Astronomical Union issue star names?
Never. The official names given to dim stars are things like TYC 3021-1882-1 or BD +40 2591. The first one means that it was mapped and measured by a satellite. The last one means that its position was first measured in the granddaddy of all massive star surveys at Bonn, Germany, in the 19th century. These are designations for the same star. The only stars that are named for people happens rarely in the literature when someone has done significant astronomical research on it, as in the fast-moving Barnard's star. Bright stars are also known by their traditional names, like Betelgeuse, or their old Bayer star map name, like Alpha Orionis, but even the companies don't try to sell these.
With solar-system objects, on the other hand, it is a different matter. Asteroids are named after people or things by their discoverers, with the approval of the IAU. Comets are generally named for their earliest independent discoverers. Moon craters are assigned names for great scientists and explorers, generally after their death. But note that you have to have significant personal involvement in astronomy or other sciences to rate such attentions. You certainly can't buy them.