A: Here's what science has to say:
For as long as there have been lights in the night sky, humans have been coming up with names for them. Sumerian astronomers named the sun, moon and five visible planets (Mercury, Venus, Mars, Jupiter and Saturn) after their great gods. In ancient China, planetary nomenclature was based on things in nature - water, fire, wood. The English names for planets mostly come from the Romans, who borrowed their designations from gods and goddesses.
Fainter and more distant celestial bodies, which can't be seen with a naked eye, generally got their titles from the people who found them.
In keeping with historic trends, those scientists typically opted for names of ancient Greek and Roman gods. English astronomer William Herschel supposedly wanted to name Uranus "Georgian Sidus" after King George III but was unsuccessful. Pluto was christened by 11-year-old Venetia Burney, a schoolgirl who suggested the name to her well-connected grandfather, who then got it approved by researchers at the Arizona observatory where the (now dwarf) planet was discovered.
We take these names for granted today, because English has become the international language of science. Most scientific journals are published in English. The International Astronomical Union (IAU), the organisation responsible for assigning designations to celestial bodies, is based in France but does its business in English. When it was established, the IAU pretty much adopted all of the English designations for objects in our solar system.
These days, the IAU has rules for naming new celestial bodies, which allow for a bit more creativity than the parameters of the major planetary designations.
Broadly, planetary nomenclature reflects the identity of the planet in question: features on Venus (named for the Roman goddess of love) are all named after women; features on the Martian moon Deimos (which is itself named for the Greek god of terror) get their designations from authors who wrote about Mars. Craters on the asteroid Gaspra are named after spas of the world. Clusters of hills or knobs on the Saturnian moon Titan are named after residents of Middle-earth.
These days, researchers on a specific mission - say, one of the Mars rovers - will compose lists of possible names that they can pull from as they discover new mountains, craters, ridges, etc. These informal names are used for initial exploration and research, then submitted to the IAU for ultimate approval. For classes of features that don't have an IAU naming scheme, scientists are free to indulge their wackiest impulses. When scientists on the Spirit Mars rover mission had to come up with a classification system for soil types, they used flavours of ice cream; now the Martian landscape is littered with rocks called "Cookies and Cream" and "Mudpie". (It's worth noting that the IAU does not approve those kinds of names - they're just for informal use.
It's extremely uncommon to name celestial bodies after people, and that honour is typically only given to prominent scientists who have died.
Likewise, no matter what anyone tries to sell you, there is no way to "buy" the name of a star. This stuff is a scam.
Stars in named constellations, which are visible to the naked eye, are typically given a letter of the Greek alphabet according to their position in the constellation. For example, the brightest star in Cygnus (the Swan) is Alpha Cygni, the next brightest is Beta Cygni, etc. Some of these stars also have names of their own (Alpha Cygni is also known as Deneb).
But there are way, way, way more stars in the universe than we could ever come up with names for. Just last week, researchers reported that there might be 2 trillion galaxies - GALAXIES! - in the observable universe, each of them containing an untold number of suns. So the IAU has settled for giving every star a number.