What if you could give it quick regen like Shadowmere? Or even invulnerability?
If I had a horse to survive 10 levels, I'd name it "Lucky".
After 20 levels, I'd name it "Darn Tootin' Lucky".
After 30 levels, I'd name it "Mage Killer".
After 40 levels, I'd name it "Invincible".
After 50 levels, it's sold for glue because I'm expecting to have a horse with its own name by then.

(huh- thought this icon was smoking)