This is a tricky situation. There are many factors that go into this. The bottom line is only one taxpayer can claim tax benefit for the child.
The grandson is the qualifying child for 2 taxpayers - the grandson's mom and the grandma. Only one taxpayer can claim tax benefit, they can't decide who gets to take which tax benefit. In this situation you either take tax benefit for the child or you don't. You both can't claim tax benefit for the same child.
If mom claimed grandson as her dependent, she gets all tax benefit and grandma cannot claim HOH. If mom is not claiming grandson, grandma should claim him and this will also allow her to claim HOH.
In IRS Pub 501, refer to page 15 under Qualifying Child of More Than One Person
Sometimes, a child meets the relationship, age, residency, support, and joint return tests to be a qualifying child of more than one person. Although the child is a qualifying child of each of these persons, generally only one person can actually treat the child as a qualifying child to take all of the following tax benefits (provided the person is eligible for each benefit).
1. The child tax credit, credit for other dependents, or additional child tax credit.
2. Head of household filing status.
3. The credit for child and dependent care expenses.
4. The exclusion from income for dependent care benefits.
5. The earned income credit.
The other person can’t take any of these benefits based on this qualifying child. In other words, you and the other person can’t agree to divide these tax benefits between you.
If there is a disagreement between grandson's mom and grandma on who will claim the child for tax benefit. The IRS Tiebreaker rule will determine it. In this case it would go to grandson's mom since she is biological parent.
There are excellent examples of situations on page 15 of Pub 501. Example 1 is this exact situation.
@lindyk
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