The main sidekick character who saves the MC life in the very beginning I just call him "The Young Man". There is alot of actions scenes where the characters don't really have time to exchange names. Then the young man is sorta pissed at the MC for not escaping, basically gives him F-you response. Eventually, the young man puts aside his blame and tells MC his name in the 3rd chapter. I want to keep the mystery a little bit for the reader on this characters name. Just reading the repetitive "The young man" is getting old. I'm like how long do I keep this going instead of beating a dead horse of using young man so much. I want to use other words like "Ginger" for him having red hair but I feel as it's become an offensive word for red hair people. I don't know.
Personally, I'd keep using "the young man". It acts like a tag, a title or label: it lets the audience know that when you say "the young man", you mean THIS GUY. Or, alternatively, have the character he's talking to (presumably your MC?) give him a nickname or label that you can use in the mean time before he gets a name. I think either of those might work well, but I prefer the former! Not to be super self-promote-y or anything, but I've got a short story in the fantasy section that does the same thing. Neither character has a name because that wasn't the focus and they never bothered to exchange them (or that was my argument, haha). It ends up just being two "labled" girls talking back and forth--one 'knight' and one 'straw-haired girl'. It seems somewhat along the lines of what you're trying to do, so maybe give it a look and see if someone else repeating a tag over and over is as annoying to you as a reader as it is to you when you write it! https://www.writingforums.org/threads/breaking-the-sky-two-magic-girls-beat-the-crap-out-of-eachother.158296/ Hope that helps in some way!
Your MC, presumably the point of view character, would probably just give him a nickname. Otherwise, disclose the name earlier. Keeping a name secret isn't mysterious. It's just annoying. What is the point of it? How does it serve your story to delay giving out his name? Is his name some twist related to the main plot like, "The lost prince everyone's been looking for!"? (I'm guessing not) Don't think yourself clever for keeping a name secret for no good reason. Your readers won't be impressed and intrigued - they'll find it annoying and pointless. Unless there's a point to it? Note: no, it doesn't add any mystery factor, trust me.
I've got a similar issue in my novel where one mc is chronically terrible with names and spends a good chunk of the early story referring to the other mc by his profession. It's not something I would normally do, but I feel it adds something because this narrator is very closed off and it's a sign of acceptance when they put in the effort to remember his name. But ... it also kinda sucks to read. The reader knows his name because there are chapters in his perspective, which I'm not sure makes it better or worse. I'm not sure if my idea adds more than the annoyance of the epithet takes away. Even if it seems a little contrived, it might be best to just find a way to introduce that name earlier for reader convenience. Have the mc call him Ginger so he can be like, "hey, fuck you, it's [name]," in one of those action sequences.
Why is this character's name a mystery? I'm not sure that's the sort of mystery a reader want, and spending a few chapters with this nameless character could turn some readers off. I am a big believer that characters should have names. Not naming them or keeping their name a secret from readers is bound to create problems like the one you're running into.
I did this. I had a character whose name was not known to the POV character, so I couldn't give his name. So I had the character notice something about him (in this case, it was that the man stood out for wearing a grey suit) and then used this as though it was the character's name.
The Ginger literally says this to the MC, I could do this and not drag it out. The 1st 2 chapters are pretty much almost sorta fast pace, adventure action scenes trying to escape. So it wasn't any time to drink some tea, be like "Hi, I'm Bob." . I figured that it was getting quite annoying for myself reading till the 4th chapter to find out his name. So I'm going to change it, maybe put it in the 2nd chapter when they're taking a breather while escaping.