I am looking for ideas on how to have my 9 year old son analyze his chess games. Is there a software that he could use? He may not want to analyze the already played games on a chess board by himself.
Beyond the fact that it's easier to go backwards and forwards through a game and try out variations on a computer than on a physical board (it remembers where the pieces were), software probably isn't very relevant because it's usually rather hard to interpret what the software is saying. Often it will highlight tactical blunders, which is fine: you look at the moves it gives you and you see that you could have won a knight or your opponent could have taken your rook.
However, computers are very poor at giving strategic guidance, which is what most beginners really need. Avoiding gross tactical blunders is mostly just a matter of practice (and, OK, it's plenty enough to win most games against other nine-year-olds). But, to get any farther, it's not enough just to play aimlessly but without blunders.
If you're an OK player yourself, you can probably offer much more to your son than any computer could. If you don't feel that you're a good enough player to teach anything much to your son, I'd say that your best options are to either both learn together (your greater intellectual maturity will help you structure things and choose what to do next; his greater ability to soak up everything like a sponge will help to push you both) or to enroll him in a chess club where somebody more experienced can take on that teaching role. If he's really keen, you could even hire a coach for him.
Thanks to JiK for a helpful comment that was edited into this answer.
You could ask him to write down his thought process for every move. If you checked books like the "zurich international chess tournament 1953" by david bronstein, you'll realize that analysis doesn't always have to be bunchy long vague variations. Simple sentences might just make sense too.
Yes, there are many softwares which do the analysis (also termed annotation) of chess games using chess engines. Some free software for Windows are:
There are numerous other softwares both free and commercial for specific platforms like Mac, Android, iPhone etc.,
Typically after your son plays the game, the game needs to be entered into the chess software as a new game. If he plays the game online, then you could use the PGN file for this purpose. The software (using the chess engine), can point out the evaluation of each move (whether its good or bad), by means of a simple scoring scheme (positive scores indicate good for white, negative indicate good for black). The GUI also point out or flags bad moves for consideration. Typically, analyzing games with the software is for finding tactical mistakes, blunder checks, ideas in specific positions and opening preparation. For a clear understanding of why a particular move was chosen as best in a position, would involve understanding positional concepts. This can be gained by acquiring knowledge through self-study or through a coach.
If software isn't working out well, you could submit the games played to a strong player or coach for his/her inputs. This facility is available online on many internet chess sites.