Which would make more sense? "How is thy morrow?" or "How is thine morrow?"

Neither really makes sense. However, "How wilt thine morrow be?" makes much more sense.

Edit:

(I was confused as to what you meant, for morrow means tomorrow. Morn means morning)

You could say How is your morning in many ways.

However, I personally like "Art thou well this morn?"

Seeing as there is no difference between thy and thine except for the sound, do what floats your boat with those.