You can only claim Head of Household (HOH) if you are providing more than half of your child's support.
The rules about only one individual in a household claiming HOH were designed to discourage parents from getting a higher standard deduction by each claiming HOH status. Since you are renting a room in your friend's home, you are technically not living in the same household.
Is your friend supporting you and your child in any way? If so, you must factor in her support when calculating how much support YOU are providing to your child. The IRS provides a worksheet
for determining support that explains what counts.
