The song is about a little girl. The song starts out with her father singing to her, "Mary, it's time to get up," to which she replies, "I don't want to get up." He then sings,"Please, please Mary, won't you get up?" She responds, rather brattily, "No, no Daddy, I won't get up!" A little while later, she wakes up with the sun shining brightly in her face. She exclaims, "What a beautiful morning, and look! There's Mr. Sun!" She then begins a dialogue song with the sun, "Good morning merry sunshine, how did you wake so soon?" The sun, quips back, "I never go to sleep, dear child, I just go 'round to see, my little children everywhere, who rise and wait for me. I've wakened all the birds and bees, and flowers on my way, and last of all the little child who stayed up late to play." Then a chorus begins, repeating the words.
Funny I should find this post, as I was just thinking about this just the other day.