Horse.
It is a good horse, and how the rider uses that horse, that makes an effecetive cavalryman; the weapons and armor of the rider are of little consequence, merely different variations of the same theme.
And honestly the number one reason for me dieing as a lancer is my courser. It's delicate, and archers love shooting gigantic targets. If heirlooming your horse gets it more HP, maneuverability, or speed--anything that increases its survivability--then I highly suggest doing that.