So, I was thinking, why can't you name your horse when you purchase it in Skyrim? I mean, the name of the Horse is already altered somehow (becoming {player's name}'s Horse) when purchased. It doesn't seem like it would be too hard to, when you purchase a horse, Script in a messagebox for user input, save into a variable and name the player's current horse said variable.

That said, I have little to no Papyrus experience. I did play around extensively with scription in the construction set for Oblivion, but with the Creation Kit I can't find exactly where the horse's name is changed at all. I can find the topic where the player purchases the horse, so I guess I could just add to the script there? I'm not even sure if user input is allowed with messageboxes, though (only user input in the game is player name and enchanting, if I'm not mistaken).

Anyone have any insight into this, if it's impossible, or if it's already been thought of, or (better yet)have an idea of going about it?

Did you look at the papyrus script section of the dialogue? there is a begin and end section, plus an actual script fragment section to the right. Any naming would have been done there. Oblivion required OBSE to do this, using the setname function, but it might be possible in vanilla Skyrim, I haven't got into any of that yet.

There is input to custom name your enchanted weapons though, so I would look there to see how they are allowing the input, and then apply that to same code to your horse.

Not possible with available papyrus functions. When SKSE come up with text input, feature you requested will be an integral part of http://skyrim.nexusmods.com/downloads/file.php?id=14950.