Have you changed your child's name?
I haven't
Is your child working currently?
yes, he is.
First, just for clarification, you should inform Social Security and the current employer of your child, if his name changes.
what could Social Security ask me in regards to his immigration status?
social security will ask you for your child's current U.S. immigration document from DHS.
is it really necessary to tell the employer about a name change?
yes, it is. If you don't, it could lead to delays in your child's and your's tax refund.
And what if I'm a foreign worker?