This miniature came to us from a disbursed collection out of Russia, and among those were several of this same child. I've looked at all the images of Napoleon's son (with Marie-Louise), who spent most of his entire life at the Schönbrunn Palace, Vienna, home of his grandmother, Empress Maria Therese (Habsburg line). This boy looks incredibly like that child. The other portraits in the group or family collection included several Habsburg line images, as well. I can't guarantee it is of Napoleon's son, but all things do point to it having been. He led a lonely and sickly short life there, dying of tuberculosis at age 21 at Schönbrunn Palace.
Very good for age and type, there is a flaw to the painting which is readily visible. A drop of water, likely, which has dissolved the paint in a spot making it almost seem like a quote bubble, holding his silence. He does hold a ball, or perhaps a piece of fruit. Might have been a quick portrait of him to send on to his mother or a family member. He was left by his mother at about age 2. A sad history. The measurements are noted on the photos. The frame and convex cover, backing - all original.