Folk etymologies here. I think Bilby's on the right lines - 'cute' was a contraction of 'acute', so originally it was another expression of bright as a button (the sense of bright having shifted from "shiny" to "smart").
My grandma often said bright as a button and I've heard it elsewhere. I sense therefore that buttons were an object of admiration for their shininess. Of course it's also possible the two sayings developed independently.