California

Girl is 21 in 2000, full-time undergrad student.

I do the mom's return. She gives me a 1098-T for her 21yo daughter: $0 in Box 1, $7,203 in Box 5.  Has maybe $K of other eligible expenses.

Mom is the custodial parent, but court order allows Dad to claim CTC (or ODC in this case).

Per the IRS ITA "Do I Have to File Tool?" the daughter doesn't have a filing requirement.

Does Mom have to claim this net ~$6,200 in income?

I could see a case for "Dad gets the ODC tax benefit, he should pick up the income" as well as "Mom's household was probably the economic beneficiary of that money."