So this idea is somewhat vaguely worded so I have a couple questions for clarification.
Where do you want to name them? Name them in our backpack or have a name other players can see?
If the name is only in our backpack, it would seem pointless to me and nothing problematic, as we can name decks. Maybe annoying to KI, as it might make it harder to store the data, but it should only be one variable added of something like ItemRenamed.
If you want other players to see your named mount (Like a chestnut pony named Elmer) I worry it might be a different question of "Where should the name render?" Should it be inside the character? Of course not! But should it replace our badge? Please no! But it will be hard, or in some cases impossible, to see it if it is above our name. And we certainly don't want it to REPLACE our name or overlap it!
The reason we have named pets and can see their names is because they are beside us and can sit nicely there. They have a model separate from us and their name is above their model. It works well. I think the problem of where to render the name is the real thing stopping mounts from being nameable.