Most working paleontologists don't usually get the opportunity to name their own dinosaur; in fact, paleontology is an anonymous and not terribly exciting occupation--the typical PhD candidate spends most of her day laboriously removing encrusted dirt from newly discovered fossils. The one chance a researcher really gets to shine is when he or she discovers--and gets to name--a new dinosaur discovery. (See a list of the 10 Strangest Dinosaur Names and the Greek Roots Used to Name Dinosaurs)
There are all sorts of ways to name extinct creatures. Some of the most famous dinosaurs are named after prominent anatomical features (e.g., Triceratops, which means "three-horned head") while other are named according to their supposed behavior (the most famous example is Oviraptor, which means "egg thief," even though the charges were overblown). A bit less imaginatively, many dinosaurs are named after the places where their fossils were unearthed--witness the Canadian Edmontosaurus.
As a general rule, dinosaurs are referred to by their genus name--the next step up the taxonomic tree from an individual species. For example, the dinosaur commonly known as Ceratosaurus comes in four different flavors: nasicornus, dentisulcatus, ingens and roechlingi. Most people can get by with just saying "Ceratosaurus," but scientists refer more precisely to the species Ceratosaurus ingens (the whole name is italicized, with only the genus name capitalized).
Dinosaur Names - "Deprecated" DinosaursAccording to the arcane (but eminently fair) rules of paleontology, a dinosaur's first official name is the one that counts. That is, if a paleontologist digs up a unique dinosaur in South America, the name he selects is the one that sticks, even if an identical species is found 500 miles away and named a week later (before the first researcher has had a chance to publicize his discovery).
Occasionally, this strict rule can lead to some confusion. The most famous example is Apatosaurus, which is now the "correct" name for the sauropod once known as Brontosaurus. It turns out that the same famous paleontologist who discovered (and named) the bones of Apatosaurus later discovered (and named) the bones of what he thought was an entirely different dinosaur. When it turned out that Brontosaurus was the same animal as Apatosaurus, official rights reverted back to the original name, leaving Brontosaurus as what paleontologists call a "deprecated" genus. (This sort of confusion doesn't only happen with dinosaurs; for example, the prehistoric horse formerly known as Eohippus now goes by the less pronounceable Hyracotherium.)
Dinosaur Names - "People"-saurs
Surprisingly, fairly few dinosaurs are named after people, perhaps because paleontology tends to be a group effort. Some legendary practitioners, though, have been honored in dinosaur form: for example, Othnielia is named after the famous paleontologist Othniel C. Marsh (the same scientist who caused the whole Apatosaurus/Brontosaurus blowup), while the dinosaur dubbed Drinker wasn't a prehistoric alcoholic, but named after the 19th-century fossil hunter (and Marsh rival) Edward Drinker Cope.
Perhaps the most widely publicized people-saur of modern times is Leaellynosaura, which was discovered by a married pair of paleontologists in Australia in 1989. They decided to name this small, gentle ornithopod after their young daughter, the first time a child had ever been honored in dinosaur form--and they repeated the trick a few years later with Timimus, an ornithomimid dinosaur named after the husband of this famous duo.
Dinosaur Names - The Big and the Silly
Every working paleontologist, it seems, harbors the secret desire to come up with a dinosaur name so impressive, so profound, and so just-plain-cool that it results in reams of media coverage. Recent years have witnessed such unforgettable examples as Tyrannotitan, Raptorex and Gigantoraptor, even if the dinosaurs involved were less impressive than you might think (Raptorex, for example, was only about the size of a full-grown person, and Gigantoraptor wasn't even a true raptor, but a plus-sized relative of Oviraptor).
Silly dinosaur names--if they're within the bounds of good taste, of course--also have their place in the hallowed halls of paleontology. Probably the most famous example is Irritator, which received its name because the paleontologist restoring its fossil was feeling, well, irritated. Recently, a paleontologist named a new dinosaur Mojoceratops (after the "mojo" in the expression "I've got my mojo working"), and let's not forget the famous Dracorex hogwartsia, which was named by pre-teen visitors to the Children's Museum of Indianapolis.