Looking in related languages
In natural languages such as English the problem is usually that not just does the language itself not have an elegant way of dealing with the problem, but there are also no dialects or other closely related languages that have a solution ready for borrowing.
With Esperanto the situation is similar but not quite the same. This problem with Esperanto's pronouns is just one among several that were addressed with Ido. As a result, a lot of Ido pronouns (all other than first and second plural) are different from those of Esperanto. Nevertheless, I think the Ido third person pan-gender pronoun lu is arguably the natural choice for Esperanto.
Apart from a general desire to keep Esperanto 'pure', even if that means protecting its most sexist 19th century features against reforms, I can see some good arguments against the use of lu, though:
- Lu doesn't fit into Esperanto's system of all personal pronouns ending in i. (This feature was intentionally changed by Ido to make the personal pronouns easier to distinguish.)
- Esperanto has a verb lui (to hire) that is of course in no way related to lu. It appears that in Ido the corresponding verb is lugar rather than luar to make absolutely sure that this cannot lead to confusion. But in Esperanto one could theoretically be misled into mis-hearing lu as luo and trying to come up with a meaning for it as a noun. (Luo is not a dictionary word, but also not unlikely as a derivation of lui. There is also the Esperanto name lo of the letter L, but I think that's less likely to be a problem.)
Nevertheless, in the sense that Ido is really just a dialect of Esperanto (I believe there are even books and plays that make use of this), Esperanto can be said to already 'have' this pronoun.
Inventing a new solution in the original spirit
One could also think of adding one that is more in the original spirit. Unfortunately I don't know any language with a gender-neutral third person singular pronoun ending in i. If we want to stick with one consonant letter plus one vowel, this leaves us a choice between bi, di, fi, gi, hi, ji, pi, ri, ti, zi, ĥi, ĵi. I think we can eliminate some of these as they are hard to pronounce, hard to distinguish from existing personal pronouns, or misleadingly close to certain pronouns or offensive words in well known languages. I would argue this leaves at most bi, di, gi, ri. I find none of them really convincing. [I changed my mind. See PS below.]
However, as I found out after writing the above paragraph, there is already a proposed Esperanto reform that would make ri a gender-neutral third person pronoun. See Wikipedia on 'riismo' for details.
Accelerating natural language change
Another option would be to just say li/ŝi quickly. Even spelled li-ŝi, this would easily be understood, and I think it would fit into Esperanto's system of pronouns. (I think it's better this way than the opposite order ŝi-li, as that would be more likely to raise ideas of something related to chili or Chile said by a new Esperanto speaker who sometimes uses incorrect nouns not ending in o.) This solution, also, is in a sense already present in Esperanto.
(Disclaimer: I know a lot about languages and have read some books and articles about Esperanto. But I am just learning to speak the language right now, so I can say nothing about how this problem is handled in practice. Basically I am just trying to bring up a few points that in my opinion are likely to play a role.)
PS: There is an extensive Wikipedia article on the problem of a seksneŭtrala homa triapersona pronomo. It contains some additional ideas that I find quite convincing: (1) Geli is arguably already an Esperanto word fit for the purpose. (2) Gi can be interpreted as either a simplification of geli or a variant of ĝi, which some people (quite logically but perhaps not very politely) already use for the same purpose. Or one could even think of the prefix ge- as derived from a gender-neutral pronoun gi, in the same way that we may say "he-cat" or "she-cat" in English. So gi seems to be an excellent fit.