There's lots of information out there saying that Fuji is a great mountain for kids to climb. And it can be. However, Fuji is a serious mountain and if Fuji is the first mountain your kid has ever climbed, you can pretty much bet that s/he will have a very hard time.
Kids rarely have physical problems when it comes to climbing Fuji. Instead, any issues they have are very likely to be mental. If your kid doesn't have much experience physically working hard for 6-8 hours a time, sweating a lot, and being outside, they are more likely to shut themselves down and not have much fun.
If they love being outside and have lots of experience doing outdoor sports, then they're likely to have more fun.
FMG has guided children as young as 6 up Mt. Fuji, but this is not the norm, this is the exception. Generally, a minimum age of 10 is a good place to start when thinking about bringing your kid on Fuji.
If you bring your kid and it isn't going well, there's always the backup plan of being dropped off at a lower hut on Mt. Fuji (with the parent), and being picked up on the way back down on day 2.