I have a character who’s a shapeshifter who changes between men and women with regularity and need a name for said character. The character pronouns themself as “they” and really has no preference concerning gender. If personality impacts it at all, they are loyal, brutally honest, highly intelligent, and care deeply for humanity as a whole. Generally I would prefer names that aren’t common because there are too many masculine/feminine connotations for popular names, but I would also like names that aren’t completely unheard of. Generally, I favour one or two syllable names.
Names that I like but aren’t right:
[name_u]Vesper[/name_u]
[name_u]Ali[/name_u]
[name_u]Emery[/name_u]
[name_u]Devon[/name_u]
[name_u]Taylor[/name_u]
[name_u]Rowan[/name_u]
[name_u]Sorrel[/name_u]
These are fairly androgynous to me:
[name_u]Denver[/name_u]
[name_u]Dallas[/name_u]
[name_u]Winter[/name_u]
[name_u]Ellis[/name_u]
[name_u]Fran[/name_u]
[name_u]Rex[/name_u]
[name_u]Lex[/name_u]
[name_u]Ivory[/name_u]
[name_u]Onyx[/name_u]
[name_u]Cass[/name_u]
[name_u]Jay[/name_u]
[name_u]Lee[/name_u]
[name_u]Cory[/name_u]
[name_u]Drew[/name_u]
[name_u]Dylan[/name_u]
[name_u]Lou[/name_u]
[name_u]Eden[/name_u]
[name_u]Pepper[/name_u]
[name_u]Dana[/name_u]
[name_u]Chandler[/name_u]
[name_u]Riley[/name_u]
[name_u]Morgan[/name_u]
[name_u]Fallon[/name_u]
Seeder
[name_m]Espen[/name_m]
[name_u]Aspen[/name_u]
Maybe you could give them a nickname, like a word or something that isn’t technically a baby name, that they could go by.
Other suggestions (sorry if I accidentally repeat pp’s suggestions):
[name_u]Roan[/name_u]
[name_u]Yarrow[/name_u]
[name_u]Jay[/name_u]/[name_u]Jai[/name_u]
[name_u]Ray[/name_u]/[name_u]Rey[/name_u]
[name_u]Ari[/name_u]
[name_u]Navy[/name_u]
[name_u]Dane[/name_u]
[name_m]Teague[/name_m]/[name_u]Teagan[/name_u]/[name_u]Tegan[/name_u]
[name_u]Arlo[/name_u]
[name_u]Dale[/name_u]
[name_u]Kai[/name_u]
[name_u]Quinn[/name_u]
[name_u]Loren[/name_u]
[name_m]Kiran[/name_m]
[name_u]Blair[/name_u]
[name_u]Remington[/name_u] (nn [name_u]Remy[/name_u])
[name_m]Espen[/name_m]
[name_m]Hal[/name_m]
[name_u]Casey[/name_u]/[name_m]Kase[/name_m]
[name_u]Ira[/name_u]
Good luck with your story!
Some of my favourite non-gendered names:
[name_u]Reid[/name_u]
[name_u]Noel[/name_u]
[name_u]Dorian[/name_u]
[name_u]Adrian[/name_u]
[name_u]Dale[/name_u]
[name_u]Quinn[/name_u]
[name_u]Selwyn[/name_u]
[name_u]Tracey[/name_u]
[name_u]Jules[/name_u]
[name_u]Dusty[/name_u]
[name_u]Brynn[/name_u]
[name_u]Kieran[/name_u]
Of course, the gendered-ness of a name is based on local context, so some of them might seem more male or female to you. But those are very neutral to me ^^
I went unintentionally nature-based. Sorry it’s a short list. Names I like that have already been mentioned are [name_u]Ari[/name_u] and [name_u]Ray[/name_u]. Good luck 
[name_u]Skye[/name_u] / [name_u]Skyler[/name_u]
[name_u]Jamie[/name_u]
[name_u]Jesse[/name_u]
[name_u]River[/name_u]
[name_u]Rain[/name_u]
[name_u]Phoenix[/name_u]
[name_u]Robin[/name_u]