I can't claim any Greek proficiency, but after looking at a number of usages, my thought is that idios is meant to describe the self or identity of a person. The lexicon describes my thought fairly well under 1.c:
Various usages are, for example:
- "His own farm" (the farm of himself)
- "His own language" (the language which describes him, his native language)
- "His own slaves" (the slaves of his, slaves by his name. While likely exclusive, that's possibly besides the point here)
- "His own blood" (the blood of Him. While obviously exclusive, that's beside the point, because it's referring to the nature of the blood being of God)
The ones translated as "themselves" and "privately" are weird, but it possibly fits there too, if understood as "personally". There's a bit of a leap there, but it could work.
Acts 4:32 is important:
At face value it sounds like idios is being used as a marker of exclusivity as opposed to inclusivity (common property). But if read as "not one of them claimed that anything belonging to him had his name on it" (as in, it was not "of him"), then it still works. In fact, it tweaks the understanding of the passage just slightly: it implies they weren't just sharing, but considered it all to belong to the community.
In this way, "each woman is to have her own husband" might be able to be understood as "each woman is to have a man to be called by", or "described by", or "be identified by".
But I'd really want someone who understands Greek to assess that.