Your friend should consult with his supervisor for sure, this seems like a pretty trivial matter to me.
Just make sure the friend wants to do this themselves, and isn't just being pressured or motivated just because some professor asked them too. I mention this because a student afraid to mention something so trivial to a PI might be one who thinks they have to say "yes" to anything someone with a PhD tells them.
If your friend wants to do this then he should tell his PI. I'd tend towards informing the PI what you plan on doing, and asking what they think rather than asking permission. You do not, and should not, need permission from anyone to write a paper about whatever you want, especially if they're not really involved.
If the PI is very against this, for some reason, then your friend should feel free to try and be persuasive. Sure, it's going to take some time, but if it's not going to tank his primary research then he should do it. Having more publications is very much in his interest, even if it's not in his PI's interest. Many grad students are too timid with their PIs. They forget that they have rights as students/scholars/employees, and often PIs will give you their opinion expecting you to push back if you disagree, and timid people won't. In fact, students who do the latter will often find their PI confused when this kind of thing is discussed. For example:
PI: "Do this by Friday."
Student: "It will take longer than that."
PI: "Oh ok next week then."
If the PI commands this student "YOU MAY NOT WRITE THIS PAPER." Then he should consider getting a new PI (if possible) because this one is terrible. If he doesn't want to do that, then consider just doing it anyway or not doing it.
I have found that many times when a PI tells you "Don't waste your time on that" and you do it anyways, they often won't seem to remember they told you not to, plus if it ends up being useful they'll be happy you did it. I often did things I was told to not waste my time on. If it turned out being a waste of time, they never found out about it, and if it didn't he'd often just remember it as his idea anyway.