I'd like to suggest 'herm' for either him or her, and herms for hers or his, or the possessive without an apostrophe.
This would be not only a logical replacement for the clumsy plural 'their' when referring to a singular someone, but lexically apt, since a herm is a) a bust of Hermes (Mercury), the god of communication, raised as a boundary marker, or b) a person with both testes and ovaries.
I know we SAY 'their(s)' and it's understood as either singular or plural, but that's colloquial. In a court of law it can be significantly misleading to say 'their(s)' when the sex and number of suspects is unknown.

In the court of law it's OK to be clumsy and awkward as long as you*'re very clear about what you* mean. In normal language invented ways of handling the singular they situation are traditionally very unsuccessful. Standard singular they, however has been with the language for a long, long time, and is generally less subject to ambiguity than singular you.

*Singular or plural you.

You quibble a problem long ago solved.
And when I say "you" I mean "you all".

What bothers me when someone uses the singular they is that they uses the plural verb form. If they wants to keep the language as clear and logical as it is now, they has to be consistent in their singular-plural inflections – you get what I'm saying? And one chao in the language could lead to chaos everywhere!

While the "you-all" strategy certainly works if it is applied universally (which it doesn't; other strategies, such as "you guys" or "yinz" are used in other areas of the country, and "you-all" has been spotted in the wild being used as a singular) it has not been accepted in Standard English.

A similar strategy to Tromboniator's 3rd person singular verb with singular "they" was used in the early days of singular "you". It did not last. The verb form has followed the pronoun and not the person and number of the pronoun.

bring back thou, thee, thine and the tribe

Really now, Faldage? If you have spotted "you all" used as a singular (as you say) "in the wild" then apparently the boys were funning with poor Mister Faldage. And -as you well know- Standard English is not standard universally, now is it?

Or either that or my sources are more widely spread than are yours, one. And, while there may be any number of Standard Englishes few, if any, recognize "you-all" or "y'all" as card-carrying members of their league.

My spies tell me there are places where "Y'all" is standard second person singular; if you mean plural, it's "Y'all all".

Now listen up, boyz, you don't need no interjecting apostrophe to indicate any missing letters when using the plural term "yall" so don't clutter up your scripts with useless marks. If the need arises to futher pluralize this plural use the term "all yall".

Sometimes I think the problem with american yankees is that you all think that all of your thoughts are worthy just because you all think that you are.

Hmm. Trying to remember the ways I heard it said. I've heard, Y'all all at the beginning of phrases, as in, "y'all all better remember to bring your bathing suits when you come over" but all y'all at the end, as in, "I brought this casserole for all y'all".

I think I like "all y'all" better!