I can't remember what is the word for those funny light spots created when you take a mirror or any reflecting surface and make a spot of light that can be moved all over the room, walls, etc. Does anyone know?
In my family (rural Australia) we have always used the term "jack-a-dandy". I can trace this back to a usage in 19th C London (sorry, reference escapes me - not my family, though.) It was an article on children's games. One boy complained that another had cast a jack-a-dandy in his eye with a mirror or similar. The trail stops there. I tracked down an earlier (?) English saying to the effect that, on a day when sun and rain alternated rapidly, "Jack-a-dandy is beating his wife with his golden stick". In Germany, in the same circumstances, it is the devil who is beating his wife. My Jamaican sister-in-law reports a similar saying in Jamaica.
Jack-a-dandy -- Wiktionary
(noun) 2. (New Zealand) A small patch of light reflected from a shiny surface onto another surface such as a wall or ceiling.
The specific optical effect you are after is:
In optics, a caustic is the envelope of light rays reflected or refracted by a curved surface or object, or the projection of that envelope of rays on another surface.
Note, that even very slightly curved surfaces such as a the glass on a watch can (and do) cause caustic reflections.