- homonym (same + name): a word that sounds like another word.
- synonym (together + name): a word that means the same as another word.
- antonym (against + name): a word that's the opposite of another word.
- pseudonym (false + name): a fictitious name for a person.
- acronym (top + name): a word formed from letters in a phrase (NASA, laser, AIDS, radar, anddepending on how broadly you want to define the word, perhaps also ETA and modem).
There are some -onym words that might be a little more obscure. An eponym (upon + name) is a word or name that's derived from a person: Alexandria (for Alexander the Great), Machiavellian, Alzheimer's disease, sandwich, boycott, silhouette. Orin Hargraves recently wrote about meronym (part + name), a word in which a part stands for the whole (heads to mean people), and metonym (after/with + name), a similar word in which one word is used to represent a related concept (the crown to mean the monarch). Even if you've never seen it before, you can probably guess that cryptonym (hidden + name) refers to a secret name or codename.
Biology gives us the tautonym ([the] same + name), a scientific name in which the genus and species are the same, like Bison bison and Gorilla gorilla. The word tautonym is also sometimes used to refer to words that are made up of repetitive sounds, like tutu, bonbon, and hubba-hubba. Speaking of language, lexicography gives us the paronym (beside + name), a word that shares a stem with another word, like wise/wisdom and female/feminine.
In geography, a toponym (place + name) is a word for a place. Examples include Seattle, which was named for a local Native American chief; France, named by the Romans after the inhabitants, the Franks; and Mauna Loa, a volcano whose name means "Long Mountain" in Hawaiian. A comparatively recent addition to the -onym club is demonym (people + name), which is a name used for the inhabitants of a particular place — Californian, Hoosier, Liverpudlian, Nazarene.
These are all useful, but some naming terms seem like they're more for fun. For example, there's contranym or contronym (against + name), which is also referred to as an autoantonym (self + against + name). This is a word with two meanings that are opposites. For example, to sanction can mean both to condemn and to permit; oversight can refer both to supervision and neglect. Fans of the Amelia Bedelia series of children's books might remember that when she was asked to dust the furniture, instead of "undusting" it, she dusted it with dusting powder — another contranym!
Another example that seems like it's just for fun is aptonym or aptronym (appropriate + name), which is a name that's particularly suited to a person. Examples are the poet William Wordsworth and the casino owner Steve Wynn. One of the running gags on the NPR program "Car Talk" was their extensive list of (fictional) aptonyms, like Complaint Line Operator Levon Hold and staff drycleaner Preston Creases. A retronym (backward + name) is a word that had to be invented to distinguish an older technology from a newer one, like acoustic guitar, analog clock, brick-and-mortar store, and snail mail.
Something that struck me about all these terms is how true they are to their classical roots. The -onym part is Greek, of course. Most of the prefixes I listed are Greek (syno-, anto-, topo-, demo-, etc.), with just a few from Latin (contra-, apt-). But we English speakers, we are nothing if not inventive with our word parts. The -onym stem is, as linguists say, "productive" — we continue to use it to create new terms. And since we are primarily familiar with English, we're happy to combine -onym with whatever word seem useful in the moment, whether it's of classical or English origin.
A wonderful English-based name is a backronym, a form of acronym where the phrase it stands for was deliberately constructed so the acronym would form a word. Commonly cited examples are SAD (seasonal affective disorder) and MADD (Mothers Against Drunk Driving). As Ben Zimmer pointed out not long ago, people who name legislative bills seem especially fond of backronyms, with resulting laws like the USA PATRIOT Act and the DREAM Act.
My current favorite among these -onyms is capitonym: words whose meanings are distinguished only by the use of capitalization, like turkey/Turkey, catholic/Catholic, and march/March. Some capitonyms even change pronunciation based on the capitalization, like august/August and polish/Polish. As listed in Wikipedia, an apparently unknown author has written some doggerel verse that highlights a few capitonymic pairs. Here's one of those poems:
An herb store owner, name of Herb,
Moved to a rainier Mount Rainier.
It would have been so nice in Nice,
And even tangier in Tangier.
After this survey of -nonymous terms, maybe you're ready for a challenge. A while back, my friends and I had some fun cataloging what people call themselves who work for a company. At Microsoft, they're Microsofties; at Amazon, they're Amazonians. Google employees call themselves Googlers, Yahoo employees are Yahoos, and Nordstrom folks are Nordies. Here's the challenge: what kind of -onym term can we invent to describe this type of name?