I've spent time over the last few days trying to develop a naming scheme for galaxies my collaborators and I have discovered as a part of one of our projects. Unforunately, it's not quite as easy as picking out family members and pets to name the galaxies after. First and foremost, it's not allowed, and second, those names hinder our research work.
The International Astronomical Union is, by agreement, the astronomical organization that officially names all objects. Most objects are designated by a catalog name and catalog number, but sometimes instead of a catalog number, the objects are numbered by there coordinates. For example, the Andromeda Galaxy is also known as M 31 (for the 31st entry in Charles Messier's catalog) and, much much much less commonly, as RX J0042.6+4115, for the ROSAT X-ray source at coordinates 00:42.6, +41:15. Our finished galaxy catalog may have 10,000 or more galaxies, far too many to name after all of our friends, family, pets, co-workers, competitors, and enemies. So, we'll be using catalog numbers, and naming the catalog after ourselves. (This is standard practice so that other people can find our papers talking about our search later, not so that we are immortalized.) As soon as the IAU approves our names, we'll be able to chug along!
There are companies that, for a fee, claim to "name a star" after you or anyone else. Many people think that this makes a cool gift, which is fine. Just be aware that these names are not "official;" they are not recognized by the International Astronomical Union. So just be aware that if you do "buy a star" to name, you do not either own the star or get to officially name it. In fact, one person bought me one of these stars, and there was no star at the coordinates they sent me! But, since I also received a star map, I was able to find the right star in a real catalog and learn its real name, which was a boring catalog number.