This is pretty simple, but requires a bit of creativity :D You're naming a horse based on its parents' names!

Player One lists two names. Player two thinks up a creative name that incorporates both of Player One's names, then adds their own second name.

You can use animal names you've already used on HP, or real life horse names, or use a name generator! Just keep it HP appropriate ;D

Example play:
Player One: Devil's Ivy, Golden Girl
Player Two: Golden Ivy, Neverland
Player Three: Evergolden, Big Chex
Player Four: Golden Chex, Where's Dancer

I'll start off with Scarlet Science and Perfect Rascal

SCARLET RASCAL, Stormy Night

Scarlet Night, First Strike 

Scarlet Strike, Perfect Anger

Perfect Strike, Twice Shy 

Strike Twice, Skippa Sally

Skippa Twice, Gray Matter

Twice Skipped Matter, Colonel Freckles

Freckled Matter, Sheza Star

SHEZA FRECKLED STAR, RUNAWAY BID

SHEZA FRECKLED RUNAWAY, Science Friction

FRECKLED FRICTION, Captain Kid

Freckled Kid, Winter Rose

KID ROSE, Chex to Cash

Cash Kid, Dancing Rover

KID ROVER, Hot Tropic

Tropic Rover, Love Letter

Letter From the Tropics, Rumble in the Deep

Deep in the Tropics, Destiny's Way

Destiny in the Tropics, The Farthest Star

Destiny's Star, Truly Devious