Saw this one when I was looking up acute from last week. ;-) If you recall, acute technically means "sharp." And so it's not great stretch for it to be applied to mental acumen as well as angles or illnesses.
What I didn't realize is that cute is a direct shortening of acute, and its first meaning, in 1731, was "clever." I've heard it used this way, but I had no idea it was the first and primary meaning.
Around 1834, American college students began taking the word and applying it to physical attributes, not just mental ones. And so cute moved from "clever" to "pretty."
So there we are at the meaning we use most today, which leads us to clever little things our kids say, like Rowyn (5) claiming, "I'm not handsome yet, Mommy. I'm still cute." ;-)