what is the name for a word that someone is known for saying? I tried looking for information on this but came up empty. I thought it was a word that ended in nym but I'm not sure.
I think "catchphrase" is the most commonly used word.
Catchphrase - A well-known sentence or phrase, especially one that is associated with a particular famous person.
Other suggestions: "hallmark", "signature [word/phrase]"
Hallmark - A distinctive feature
I don't know a -nym word, unless it's a neologism.
One term that fits the definition of "a word that someone is known for saying" is pet phrase. In ThoughtCo.'s Glossary of grammatical and rhetorical terms pet phrase is defined as follows:
Pet phrase is an informal term for an expression frequently used by an individual in speech and/or writing.
A pet phrase may be widely known (a cliché, for instance) or peculiar to the individual who employs it.
The ThoughtCo. page includes authentic examples of pet phrase in various contexts. For example:
"Whenever his conscience pricked him too keenly he would endeavor to hearten himself with his pet phrase, 'All in a lifetime.' Thinking over things quite alone in his easy-chair, he would sometimes rise up with these words on his lips, and smile sheepishly as he did so. Conscience was not by any means dead in him." (Theodore Dreiser, Jennie Gerhardt, 1911)
Using Pet Phrases in Writing
"Repeat a distinctive thought or phrase of dialogue in the story. This connects an earlier part of the story to a later one without having to rely on an overt transitional device. Television shows frequently overuse this technique, giving one character a pet phrase that he repeats ad nauseam. One way to vary the device is to give it a different meaning each time it's used. On Seinfeld, all the principal characters would use the same phrase, often with a different meaning, all in the same scene, creating a device all its own." (James V. Smith, Jr., The Writer's Little Helper: Everything You Need to Know to Write Better and Get Published. Writer's Digest Books, 2012)