Plural possessive with separate possessions

Solution 1:

The construct you suggest is rather awkward as you said yourself. The best way to talk about the specific house is to rephrase the sentence and say instead: "...this will probably entail a party at the house of one of the families."

Solution 2:

We can say "one of the families' houses" to refer to one of the houses owned by one or more of the families. In this context, "one of" applies to "the families' houses"

You actually could have stopped right there because that IS the correct way to punctuate the particular word order you want to use. It's true that you could avoid the issue by choosing a different way to say it, but "one of the families' houses" is also fine.

The use of the phrase "one of the" seems to apply to both "families" and "houses" but from a structural standpoint it actually applies to "houses" in this particular phrase, so all you need to do is put an apostrophe at the end of "families" to make it correct. In this case the possesive form of "families" functions like an adjective (even though it's not - there's some other term for this) describing one of the houses. No matter how many houses there are, it is still going to be ONE of them, and adding "families'" in front makes it more slightly more specific.