Usually it's pretty clear when we need to name our characters. Main characters, and the people they primarily interact with, should have names. Even if you are writing in first person point of view, it's a good idea to give your character a name by putting it in the mouth of a secondary character early on. Knowing a character's name gives us a greater sense of grounding.
In the story I'm currently working on, there is a very important character. She is something of an antagonist, and the leader of a group of people who are acting as pirates and kidnappers. Because my protagonist becomes her prisoner, she has no need to give her name. And so she didn't. I used the technique that is most common in these cases, to use some physical attribute to describe her. She was "the scarred female."
So far, so good. However, she ends up aligning herself with the protagonist at a certain point, and they travel into the city together. At this point it would not be hard to give her a name. Actually, it would be far better! Once she and the protagonist are acting together, to call her by a phrase like "the scarred female" becomes really long and awkward.
Except that her name is not in a spoken language. It's in a sign language.
I tried at first not to name her at all, but the long-and-awkward thing was really bad. The number of times you have to mention a character who is acting alongside your protagonist is high enough that not naming them is a real problem.
I had a real problem with the idea of giving her a name in the main alien language, though, because the sign language in this case is a language of rebellion, and to call her in the main alien language would be to name her in the language of the people who enslaved and scarred her.
In the end, I made a decision. She needed a name, and I would give it to her in the main alien language, but I would mark it as problematic. She introduces herself like this:
"I have been known as Othua."
At first, my protagonist doesn't realize how problematic this name is, but I have decided that before I got to the end of the story, I would make my protagonist learn the truth about what that slave name means to her. And at that point my protagonist would stop calling her Othua and learn her real name. That real name would be described gesturally and its semantic content - Silent-Speaker - would be used thereafter.
I found it very interesting to work through the complexities of being true to my worldbuilding while handling this name issue, so I thought I would share.