Probably not, but it depends.
First, he can't file HOH unless he provides care in his home for another qualifying person, usually his child dependent (although it could be a younger sibling if you don't claim the sibling as a dependent. It could also be a parent if your income is less than $5050 and you don't file a joint return). You don't mention a grandchild--no child dependent, no HOH.
Then, he might be able to file as HOH if he provides more than half the cost of maintaining his household. A household is not necessarily the same as the people living under one roof, but it can get complicated. I saw the following example somewhere.
A single mother with child lives in a 3 bedroom apartment. She rents one room to a friend who has a child. The friend pays fair market value rent, maintains separate food, and the families don't play together or interact other than as roommates. They have separate households even though they live together, and both mothers can be heads of their respective households.
Or, a single mother with child lives in a 3 bedroom apartment. She allows her sister to stay in the 3rd bedroom with her child. The sisters share expenses, share food, share chores, and the kids play together and generally act as a family. There is only one household here, and the only adult who can claim HOH is the one who pays more than half the total expenses.
So he could claim HOH if you have one combined family household and he pays more than half the total expenses for all of you, or he could claim HOH if he maintains a separate household under your roof, which you would have to prove to the IRS if audited.
There can be cases where a grown child takes care of their parents, pays more than half the cost, and can use their parents to file as HOH. It doesn't sound like that is the case, because your income would have to be so low that you don't file a tax return. Unless your son cares for a child dependent AND you can show your households are separate and not a blended family, he files as single.
