Pet peeves often fit into one or both of two categories. Category one is the most common and includes things like being annoyed at people who never turn off their turn signals or people who use the work "like" before everything they say. For example: "Like, I really hate it when people like forget to turn off their signal after they've like made their turn."
The second category, however, tends to include more obscure annoyances like being annoyed by people who prefer to play only Dungeons & Dragons 4th edition or being annoyed by people who prefer to eat their food with their mouths closed.
"There's too much silence in here while you're eating! Please give me more smacking of the lips while you eat your chocolate pudding!"
Why something is a pet peeve to your character, however, can often be just as fun to create as the what. Why your character is annoyed by people who don't smack their lips when they eat could hark back to a childhood cultural influence where it was customary to do so; and not doing so is interpreted as offensive to the chef. But because your character didn't grow up in that culture, the smacking of lips may just sound disgusting or your character might even have misophonia.
*Merriam-Webster Dictionary: a condition in which one or more common sounds (such as the ticking of a clock, the hum of a fluorescent light, or the chewing or breathing of another person) cause an atypical emotional response (such as disgust, distress, panic, or anger) in the affected person hearing the sound.
Whereas using the word "like" could actually be the formal (as opposed to the informal) way of speaking when in the presence of royalty. But because your character is unfamiliar with that custom, they may just see it as a lack of intelligence or communication skills.
How your character handles or reacts to their pet peeve(s) can evoke interactions that span from "tear-inducingly funny" to "cause for retaliation offensive." Discuss your character's pet peeves with your DM/GM and your fellow players to help ensure that no one at the table is offended by the target of the pet peeve and/or try to figure out how to incorporate them into the game without turning your game nights into a hostile environment.
For example, if your character is annoyed (pet peeve) by people who are LGBT+, work with your fellow tablemates to see how this annoyance can become a learning moment for your character's growth without becoming offensive to everyone else.
*Merriam-Webster: Misophonia (https://www.merriam-webster.com/dictionary/misophonia)