I am looking for a word or phrase to denote such a unique phrase that somebody habitually says. This phrase could for example fill the following gap:
My buddy Bill always exclaims “oh wow” whenever he is surprised. “Oh wow” is a _____ of his.
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This can be a catchphrase. I think this is most commonly applied to fictional characters, who often have very distinction catchphrases (such as Sheldon's "bazinga" on The Big Bang Theory), but it could conceivably apply to actual people if they have similar ideosyncracies.
How about signature saying or signature phrase?
For example, here are several signature sayings/phrases used by President Barack Obama:
“Let me be clear”
“Make no mistake”
“Here’s the deal”
“It will not be easy”
One could also call these catch phrases, but the addition of signature ties them to the person who uses them habitually.
signature: a distinctive pattern, product, or characteristic by which someone or something can be identified
My buddy Bill always exclaims “oh wow” whenever he is surprised. “Oh wow” is a signature saying/phase of his.