Son drove Postmates, made $606 in 2020. His total income is still under $12,400. However, the program says he owes tax because he has "business income". Is this right?

Yes, his income from Postmates is considered "business income" (i.e. self-employment) and is subject to self-employment tax, even if he didn't earn enough to owe any income tax.

He can reduce his self-employment income tax by deducting the costs he incurred to generate that income. The most significant cost would be the mileage he put on his vehicle while making deliveries.

TurboTax Self-employed (or Home & Business) has the interviews and forms needed to identify and report those costs. 

if his schedule C net income is over $400., he will owe self-employment tax of about 15% on the net. 

Thanks, this helps a lot!

Thanks, this helps a lot!