Does my child file their own taxes if they are a dependent on mine?

Yes, they still file BUT have to choose "Yes: Can be claimed as a dependent".  They won't get as much back due to that.  Next year, he can choose that he cannot be claimed if he is no longer dependent on you. 

For example, I was still being claimed as dependent until I moved out/left school.  I still filed my taxes and had to amend my return one year because my mother did not end up claiming me that year, like I thought she had, and I received $400 back because of it.  So just let him know that he must claim that he is a dependent and be on the same page next year as well. 

If he provided more than half of his own support, he would not qualify as a dependent.