She babysits my son in her home and I paid less than $3,000. It is treated as more of a grandparent-type relationship (i.e. flexible hours, sometimes overnight, and weekends). Would it be better for me claim it as a gift rather than child care? If I claim it as childcare does she claim it as 'other income' on her joint return with my dad or self-employment income?
The money you paid her depends on the intention behind it. If it were not your relative, payments for child care generally be considered self-employment income. However, if out of the kindness of her heart she sat with them, and out of the generosity of yours, you gave her a gift, then that is what it is.
If it is a gift, you can't claim the credit.
But, if you claim the dependent care (day care) credit, she must report it as Self Employment, not "other income". It will be subject to self employment (social security and medicare) tax, as well as income tax