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You prevent writing a comic-relief character by making them essential to the mission. And not shallow or dumb.

Give him a skill (besides punning) that the others value, or even a trait: Perhaps he likes punning because he is also extremely inventive in a good way, for solving problems, for anticipating the enemy, for thinking around corners. Maybe puns, bad and good, are just signs of an always active mind looking at everything from six sides at once.

So his friends put up with the puns, because his insights and strategies save their butt more often than not, and they can actually be grateful to have him. Even if they don't find his puns funny, the gratitude they have for his contributions far outweigh any irritation they have at dumb puns; so they don't resent them. They just ignore 90% of them, and laugh at 10% of them. And he can be fine with that, he made them laugh.

To give two examples to @Amadeus' excellent answer, take a look at the characters of Marco from Animorphs and Sokka from Avatar: The Last Airbender. Both were comic relief characters on their face, but after spending time with them, we quickly see that they are the tactical masterminds of their own teams.

Sokka was the planner and could easily assess their strengths and weaknesses and think outside of the box. As much as he was the funny guy, he was also the idea guy and could easily see the solution everyone was missing. Marco's humor was more of a sarcasm response to the plan that pointed out the very big flaw of the mission of the book... he rarely actually devised the solution, but once one was put forward, he pointed out flaws in a form that the team could accept. The team leader, Jake, was his best friend and had long before the series started accepted the fact that "Marco is going to mess with you... just accept it and be prepared to counter it". In fact, Marco is the whole reason Jake was declared leader... he was the only person on the team everyone had a pre-existing relationship with and he could translate Marco for everyone else.

Both of these characters allowed their prefered humor to inform their useful skills.

Another characteristic of both of them, is that they tended to not so much say and do funny things, but react to bizarre stuff going on around them. Both characters do observe that their teams tend to get into bizarre situations all the freakin' time, even by the setting's logics and are quick to point it out... and get frustrated when the logic doesn't work.

For me, the annoying thing about "funny" characters is where that the humour gets injected artificially, and for some reason they decide they have to tell a joke out of nowhere. That's great for a stand-up routine, but it doesn't make for a good story. In real life, what you want to be emulating is not being a comedian but being witty. Often this has a large element of snark included as well.

Tyrion Lannister is a great example of this. Even when he's arguing for his life, he's witty.

Tyrion Lannister : Wait. WAIT! Wait, wait! Wait, you can't just hand a dried cock to a merchant and expect him to pay for it! He has to know it came from a dwarf! And how could he know unless he sees the dwarf?

Slaver : It will be a dwarf-sized cock.

Tyrion Lannister : Guess again!

Or the interplay between Jean Tannen and Locke Lamora.

One of the odder services the Villa Candessa provided for its long-term guests was its “likeness cakes”—little frosted simulacra fashioned after the guests by the inn’s Camorr-trained pastry sculptor. On a silver tray beside the looking glass, a little sweetbread Locke (with raisin eyes and almond-butter blond hair) sat beside a rounder Jean with dark chocolate hair and beard. The baked Jean’s legs were already missing. A few moments later, Jean was brushing the last buttery crumbs from the front of his coat.

“Alas, poor Locke and Jean.”

“They died of consumption,” said Locke.

Wit flows naturally from the situation. For all normal mortals, it's hard to be witty, and we often only think of the best comeback an hour later. Your characters are in luck though - they (you) can think about it for as long as it takes. Sometimes we'll also chuck in a comment we thought was witty, but turns out to be a total mood-killer. Your characters are in luck again - they can unsay those comments if it turns out not to work.

Scott Lynch's writing of the Gentlemen Bastards is exquisite for this, by the way. Of course they're conmen so by definition they're quick-witted. But Lynch's writing doesn't just set up for a Terry Pratchett one-liner. He's writing a small group of close friends who are all highly intelligent, so one witticism sparks another character saying something, and so on, as actual conversations do. Oh, and he can probably out-grimdark your grimdark...

The question is are you writing a point of view character? If it's point of view then you have to be funny. It's the same way as you would struggle to write a millinery tactician POV character if you did not know anything about military tactics. You just can't sell a character's thoughts to the reader if you don't have a good understanding of the subject matter he is thinking about. Humor is no different.

If you are not writing a POV character then everything is much easier. Arthur Conan Doyle did not have to be as smart as Sherlock Holmes because the POV character was Watson. In the same way you don't have to be funny if other characters are hearing the jokes.

This reminds me a lot of a character in one of my works that talked too much. I did not bore the reader with text to make them pick up on that trait. Instead the POV character said things like "He said something about his dog. I tuned out after the second sentence. His lips continued to flap for the next fifteen minutes"

You can do a very similar thing for your writing. "Someone brought up the President and Bob had to tell a joke. I rolled my eyes before he even got to the punchline." Simple characterization of Bob and the speaker, and no humor had to be written.

You could claim that you can use the same technique with POV. "I told a joke, but no one laughed" but I don't think that works well. With a POV character this seems to quickly take you from immersive writing, to descriptive writing, and that is never a good thing.

Just write it and trust to both the character and situation to keep it organic.

I have a character in my current work who has quite a dark sense of humour. He is a seasoned CIA officer who has seen and done much, but because of this his wry wit evolved. I have a scene where my MC - hunted by others - is essentially trapped by his loved ones and receiving medical treatment.This character finds that highly ironic and begins to smile. As time passes, the smile becomes larger as the absurdity of the situation is savoured by him. The MC and he exchange glances and he starts laughing. The MC joins in, seeing the ridiculous side to the situation he is in.

A third character refers to this CIA officer no longer as ghost (he does not know who he is, but what is clear) but chuckles and the monicker sticks.

As long as it is natural to both character and story, you should be fine. You say it is a trait, as long as it is one facet among others you will have a well rounded character whose point of view is such that humour is added. He can give comic relief without being comic relief.

An alternate approach to the ones suggested so far:

It's a coping mechanism. If the tale is grim in general, it wouldn't be out of place for a character who has seen some stuff to cope by making jokes. In this case, he's not doing it to make others laugh (not that that's an issue if they do), but to keep his mind off of what he doesn't want to think about. So: if he makes a joke nobody else catches, he might smile but then just move on. If he's caught thinking, he responds by making a joke instead of answering seriously.

Granted, this might be a major shift and not possible for your character.

An example of this is, perhaps, Cody from Brandon Sanderson's excellent series The Reckoners.

Comic relief characters are actually more of a tool to use when the writer(s) need to break tension, sadness, or emotion from certain moments. What makes a character a comic relief is that their humour is focused on such events. So, if you want the character's humour to feel natural:

1. Make it part of his normal self:
   Dont't just have him crack jokes and puns on such strong moments; he has to react and respond with humour almost everytime specially on situations that don't affect itself directly.




2. Make his participations more than just making jokes: the character has important things to do for the plot; adds something important to the story; is useful to the rest of the characters; it just does all of that while making jokes and puns. If the humour is only part of what the character does and not its main focus, it will feel more natural.




3. Be sure that the reader knows that the character's humour isn't always funny for everyone -- including the character itself: Just as you said, not everyone likes the same things. But not only that: a person like your character knows that their jokes are not always funny, even sometimes for themself; but still makes jokes because it's their thing. Just describe how sometimes the other characters roll their eyes with the jokes; show how the character sometimes makes a pun without waiting for reaction because it's not fun for themself. This will help to make the humour inherent to the character and not to the moment.




4. Give the trait a backstory: A trait is a natural part of a character if there's a reason to it. You say you have a grim cyberpunk environment; sometimes humour is a defense mechanism to a traumatic experience, so use that in favor of the character. You can even setup a moment when some character (the MC perhaps?) is getting tired of the jokes and puns, and someone else who knows this particular character from long ago can explain why is it like that, when did that trait appeared, what experiences made the character what it is now. This also helps the readers to connect momre with the character, specially if they're also getting tired of the jokes and puns.

The point is to make the humour of the character only one of the reasons the character exists; with that it will be more than just a comic relief used to break tension.