My 19 year old child is a full time college student. He earned $5000 last year. Does he have to file a return?

No, he does not have to file a tax return.

Your child is still your dependent. He is not required to file a tax return as he has no taxable income provided the $5,000 is his only income. He may need to file a return if there is income tax withheld from his paychecks and it is worth the trouble to claim the refund (not spending $100 to claim a $5 refund).