"Head of Household
You may be able to file as head of household if you meet all of the following requirements.
1. You are unmarried or considered unmarried on the last day of the year. See Marital Status, earlier, and Considered Unmarried, later.
2. You paid more than half of the cost of keeping up a home for the year.
3. A qualifying person lived with you in the home for more than half the year (except for temporary absences, such as school). However, if the qualifying person is your dependent parent, he or she doesn't have to live with you. See Special rule for parent, later, under Qualifying Person." See IRS Pub 17, page 23.
According to you, the son does not live with the taxpayer AND does not have an ITIN, so the taxpayer cannot claim head of household, lacking a qualifying person. Please see Table 2-1 on page 25 on the link above (Pub 17).
"If the fact that the son lives with someone who doesn't file US taxes makes him a Qualifying Person,"
I do not understand this statement. Is it what you meant to say?
@PhylI
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