Do I have to claim if I made less than $300 dollars, 19 and considered dependent?

I live in Illinois.

My son worked last year but only earned $258.64.  They withheld $10.45 in Federal and $0 in state. Does he need to file a separate tax for himself.

He can file a return to get the $10.45 back. His return must say he can be claimed as a dependent on someone else’s return.

You are not required to file a tax return for earnings of less than $300.  If any taxes were withheld (doubtful) then you could file for a refund.  You would not get back anything withheld for Social Security or Medicare.