Yes, if none of the distribution had been used for qualified expenses, $13,500 (the earnings) would be the taxable amount.
Based on the numbers you provided, you have $14,000 of qualified expenses for the 1099-Q ($500 of tuition not covered by scholarship + $13,500 room & board).
14,000 / 24,000 = 58.333% of the earnings are tax free; 41.667% are taxable
She has 0.41667 x 13,500 = $5625 of taxable income to report and TurboTax should calculate that. This will be subject to the "kiddie tax".
Books and computers can also be added to qualified expenses.
You have no apparent expenses left over for claiming the American Opportunity Credit, on your return. But see below. In fact, if she has no other income to report, she should claim up to $10,000 of the tuition for the 1099-Q, declaring the scholarship as taxable, rather than the 529 earnings. This is because taxable scholarship is treated as earned income for purposes of a dependent's standard deduction (earned income +$350) and the "kiddie tax" will not kick in. Her standard deduction wipes out all taxable income. She pays no tax. When a dependent, under 24, has no earned income, her standard deduction is only $1100.
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There is a tax “loop hole” available. The student reports all his scholarship, up to the amount needed to claim the American Opportunity Credit (AOC), as income on his return. That way, the parents (or himself, if he is not a dependent) can claim the tuition credit on their return. They can do this because that much tuition was no longer paid by "tax free" scholarship. You cannot do this if the school’s billing statement specifically shows the scholarships being applied to tuition or if the conditions of the grant are that it be used to pay for qualified expenses.
Using an example: Student has $10,000 in box 5 of the 1098-T and $8000 in box 1. At first glance he/she has $2000 of taxable income and nobody can claim the American opportunity credit. But if she reports $6000 as income on her return, the parents can claim $4000 of qualified expenses on their return.
Books and computers are also qualifying expenses for the AOC. So, extending the example, the student had another $1000 in expenses for those course materials, paid out of pocket, she would only need to report $5000 of taxable scholarship income, instead of $6000.
