Sarada and I talked a bit about pronouns yesterday and tentatively agreed that a possible gender-neutral option was "mer". Another one which has been bandied about by people I don't know was "shim". Both of these, right now, are just bald words which might be pronouns but miss a particular feature of English pronouns to which I am very attached, namely case inflection.
It's sometimes said that English lacks case. This isn't true. All languages have cases, but they don't all express them by changing the words. In English they can usually be replaced by prepositions such as "of", "to" and "with". German is a familiar language which preserves a fairly accurate relic of how English used to be, in terms of case among other things. One of the adult ed classes in German I attended was remarkable for the fact that when we hit the subject of case inflections, known as declensions, most of the students stopped coming because they found it too daunting. When I tried to learn German at school, I found the dative case in particular very hard to grasp. In fact, German is pretty straightforward with regard to declension compared to some languages, since they can have up to about thirty compared to the four of German. Incidentally, this isn't as bad as it sounds because on the whole in a lot of languages they just amount to preposition-like endings stuck onto the ends of other words.
Back to English though. Before the Norman conquest, English had five declined cases: nominative, accusative, genitive, dative and instrumental. I like the name of the last one because it makes it sound like you just stop talking while a brief snatch of music plays instead of you saying the word, but that's not actually what happens. English still has a declension of sorts although it now contains at most three cases: nominative, oblique and genitive. It might become a bit clearer if I call the genitive the "possessive", which is what that in fact is. Nowadays, nominative and oblique nouns are exactly the same in form, but pronouns are another matter. This is how they look in present day English, including "thou":
English pronouns carry with them the relics of the old case system. For instance, only the objectives end in "-m": whom, him, them, 'em ("'em" is actually a separate pronoun and a relic of the true English pronoun which was replaced by the Old Norse "they"/"them", not an abbreviation of "them"), and "-r" in a possessive implies plurality and/or femininity. These features were more widespread in the past and some even affected nouns.
The dative and accusative merged long ago, although the dative form of the pronouns survived rather than the accusative, so it's "whom" rather than "whone" and "him" rather than "hine" for example. Another form which survived was "why", which is the instrumental of "what", the original having been "wherefore" but the sense has shifted. "Why" used to have a parallel in "thy" for "that", but the more recent "thy" is unconnected. The older "thy" however, is preserved in "the more the merrier", where the word "the" was originally the instrumental of "that".
Why am I going on about this? Well, it's about the possibility of a new ungendered pronoun, you see. For it to work, it has to fit into the system or it will feel unnatural, like "chairperson" instead of "chair", or, sadly "Ms" instead of "Miss/Mrs", which is particularly unfortunate. No nominative personal pronoun in English has an R at the end although Old Norse, which gave us "they", does have such pronouns, for instance "they" itself, which was originally "þær"/"þau"/"þeir" in the nominative. So I suppose I could imagine that "mer" originally came from Old Norse and unlike the others retained its "-r". The Old Norse word for "two" also ends in an R in the nominative, whose feminine is "tvær", and the rest goes "tvær" again (accusative), "tveggja" (possessive or genitive) and "tveim". All of those are decidedly un-English but would adapt to English easily: "twain" is from the English "twegen", which is accusative. "Tvær" is used to refer to "they two", so it is even used as a personal pronoun. Another thought I had was "ear", which used to be "eare" and becomes "earen" in the genitive and dative.
So what does that give me? Well, one option seems to be "mer - mer - meren - meren" like "ear" used to be. Another is to imagine it's borrowed from Old Norse and to go with something like "mer - mer - meya - mem" or "mer - mer - mera - mem". However, complications would also lead either to levelling or it just not being used and I'm over-complicating and engaging in mind games again.
If it's going to be "mer" I suggest either "mer - mer - mers - mer" or "mer" all the way through, even for the genitive.
There's today's for you then, and I realise I've been all round the houses to come up with something really simple which any normal person would've come up with immediately, but that's my brain for you isn't it?