English actually has a gender-neutral third-person singular pronoun: it’s “they”. Unfortunately, there is a crowd that will do everything they can to keep you from using “they” as anything but a plural. (They won’t catch every singular-they, of course, because it’s such a natural part of English that it often slips past them.) A ripple effect of their position is the periodic attempt to introduce new gender-neutral third-person singular pronouns such as “ze”. These are unfortunate because they add extra effort for both writer and reader when the use of “they” would be clear and simple if not for the constant threat of these pedants.
This same crowd will often decry the state of writing today with the spelling of “you” as “u” being one of their main bugaboos. I personally never spell “you” as “u”, even while texting, but surely one can see the basic logic behind it: the singular first-person pronoun pronounced like the letter “I” is in fact spelled with the single letter “I”; why shouldn’t the singular second-person pronoun pronounced like the letter “U” be spelled with the single letter “U”? This makes so much sense that I almost have to wonder about the basic sanity of anyone that thinks it’s a problem.
Anyway, here is my own proposal for a singular gender-neutral third-person pronoun. “I” is a singular gender-neutral first-person pronoun. “U” is a singular gender-neutral second-person pronoun. So let’s make “E”, derived from the common vowel sound in “he” and “she”, the singular gender-neutral third-person pronoun. We’ve solved a supposed problem and at the same time made English more consistent and logical. Win-win for everybody.