What's the deal with other companions really liking, or even really disliking new names?

[Godwin]

It's a bit weird.. does it matter at all?

This is a message when you rename someone.

[ТЕХНОВИКИНГУ]

good question. im afraid to continue renaming them :D

[Stone]

I can't see any mechanical issue with this. If there is some companion status malus in the background, I do not know.

[Valkynaz]

No clue, but I can't tell if there's actually any difference, might be like the old titles?

[Godwin]

I after posting this did some testing: It's random.
And as far as I see there's 0 consequences, just flavor. But if you want to be sure, just do the following:

Rename -> Jim
"Companions dislike"
Oh, then:
Rename -> Jom
"Companions w/e"
Rename -> Jim
"Companions like the name"

If they still don't like it at the 3rd renaming, just repeat the above steps until they do, this is how I found out it's random.. there's no memory of which name they liked or not.