I would like to see a completely new mount system, though I have a feeling that most people would hate it.
I would love to see a mount level system and mount families added, similar to the battlepets.
Both for flying and ground mounts.
First all flying mounts would have to have the dragon flight mechanics, and within your flying mounts would have families like
Examples.
Dragons: basicly being like what we have seen in dragonflight.
Gryphons: sharper turning, more acrobatic / agile. Flap - Gain speed and altitude.
magical: (flying carpets, discs and books) Has a manabar for flying, blink abillity to gain altitude.
Mechs: nitro boosts for extreeme speeds for a short while, hover (for a short time)
So on and so forth, then for your character you could level up your riding skill with certain families to make you better at your favorite mounts.
Same thing I think ground mounts should have, families and different abillities.
Horses: Kick, can deliver a powerful kick knocking enemies back a short distance if target is behind you.
Wolves: Savage charge, charges towards a target. 30 yard range.
cats: short prowl, mounted stealth, while stealthed only move 60% mountspeed.
Raptors: raptor jump, long jumps
I think all this would be really cool then for me, I would also love that your racial mounts gave you slight bonuses. as if you are a blood elf and ride a hawkstrider you leveld 10 or maybe 15% faster in your strider abillities, and maybe had like a 3% speed increase in your racial mount speed. Something nice but not so if you were a night elf and didnt ride a cat you feel disadvantaged.
I also would like to see small racing events, both for ground and for flying.
vanilla thousand needles salt plains kinda thing, for cosmetic rewards and just fun little mini games.
You also have the Argent tournament mechanics you could re use, have lancing events and other tournies around the world. a small pvp aspect to it.
That way people could work up their mounts, get more attached to them and movement would have a bit of gameplay too it.
I think that would be fun. 
Wish ya all a great day.