I believe several aspects of this answer are wrong. See my answer below and read Publication 970. In some (perhaps even most) cases, the right way to handle this is for the student to report the taxable scholarship income, while the parent files form 8863 and claims the credit. Also, a different limit applies to taxable scholarships, not $6100 or $6300, as they are unearned income.
Edited - Per following discussion, it seems taxable scholarships are earned income for reporting requirements of a dependent child, and unearned income in some other situations.
