Let's say a POTUS is elected who lives with a friend. They're basically family, whether they be his girlfriend, or just his bro, or whatever. Would they be allowed to live with them as part of the "first family"? Or must they be related more or less?
During the Bill Clinton administration, there were allegations that he was selling access to the Lincoln Bedroom. So he released a list of visitors:
The release lists 404 “family members, friends and supporters, public officials and others” who have spent the night at the White House or the Camp David presidential retreat in Maryland since Mrs. Clinton effectively launched her campaign last July in New York.
Making the List
The list is broken into five categories: “Arkansas Friends,” “Longtime Friends,” “Officials and Dignitaries,” “Arts and Letters and Sports,” and “Friends and Supporters.”
So there don't seem to be any limitations.
It's possible that the Secret Service may have some ability to reject visitors or residents for security reasons. But they don't publish any criteria on that.