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.