No, he can't claim your son if they're not married. No, she can't claim him if she has no income.
Did you get a notification that someone else had already claimed your son? What usually happens in those instances is that the IRS will ask for both parties that claimed the child to send documentation verifying their qualifying relationship to the child and residency for the entire year. Of course, only ONE of the parties will be able to give all of the information as requested.
If the program is preventing you from e-filing, go ahead and print out your return and then mail it in. The IRS will be in touch. If you are able to e-file, go ahead and e-file, and if your son was indeed claimed by someone else (or is claimed before tax season is done) they will be in touch. They will not, by law, be able to tell you who else claimed the child.