Solved
My child (13 years old) earned income from the following sources in 2016: babysitting, leading youth groups at our synagogue
Yes, the child must file a tax return because the child will owe self-employment tax even if the child does not owe ordinary income tax. 1099-MISC income reported in Box 7 and "babysitting" paid by cash or check is considered by the IRS to be self-employment income. It doesn't matter whether the child has a formal business or not. That is not relevant at all.
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