I think that my partner might be considering changing their name, although I’m not sure. She is gender-questioning (is AFAB & currently identifying with their birth sex but presenting as non-binary) & wishes their name was more gender-neutral. One day, me being me, I asked her what her name would be if she chose it & they said [name_u]Sam[/name_u], only because it was unisex.
Either way, I AM curious about this, unrelated to my girlfriend. 
I know this isn’t what you mean but there are names like [name_u]Sam[/name_u], [name_u]Alex[/name_u], and [name_u]Charlie[/name_u]. Then there are ones like [name_u]True[/name_u], [name_u]River[/name_u], [name_u]Banjo[/name_u]. Id go with a word name for something gender neutral since the others are just gendered both ways. 
My favourite unisex names:
[name_u]Ariel[/name_u]
[name_u]Aster[/name_u]
[name_u]Emerson[/name_u]
[name_u]Ezri[/name_u]
[name_u]Finley[/name_u]
[name_u]Grey[/name_u]
[name_u]Hadley[/name_u]
[name_u]Haven[/name_u]
[name_u]Hayden[/name_u]
[name_u]Jamie[/name_u]
[name_m]Jasiel[/name_m]
[name_u]Jules[/name_u]
[name_u]Katriel[/name_u]
[name_m]Maxence[/name_m]
[name_u]October[/name_u]
[name_u]Paris[/name_u]
[name_u]Phoenix[/name_u]
[name_u]Riley[/name_u]
[name_u]River[/name_u]
[name_u]Rowan[/name_u]
[name_u]Quinn[/name_u]
[name_u]Sage[/name_u]
[name_u]Sailor[/name_u]
[name_u]Skyler[/name_u] 
[name_u]Spencer[/name_u]
[name_u]Winter[/name_u]
[name_u]Wren[/name_u]
I think the most gender neutral are nickname-y names and some word names. It depends on the popularity in your area, though.
Alex
[name_u]Terry[/name_u]
[name_u]Sam[/name_u]
[name_u]Kit[/name_u]
[name_u]Quinn[/name_u]
[name_u]Avery[/name_u]
[name_u]Cal[/name_u]
[name_u]Cameron[/name_u]
[name_u]Jordan[/name_u]
[name_u]Fran[/name_u]
From my list:
Claude
[name_u]Kit[/name_u]
[name_u]Bellamy[/name_u]
[name_u]Eden[/name_u]
Léonide
[name_u]Penrose[/name_u]
[name_u]Marlowe[/name_u]
[name_u]Adair[/name_u]
[name_u]Auden[/name_u]
[name_u]Honor[/name_u]
[name_u]Harvest[/name_u]
[name_u]Sage[/name_u]
[name_u]Laurie[/name_u]
Atlas
[name_u]August[/name_u]
[name_u]Brooklyn[/name_u]
[name_u]Brooks[/name_u]
[name_u]Clarke[/name_u]
[name_u]Darcy[/name_u]
[name_u]Eden[/name_u]
[name_u]Elliott[/name_u]
[name_u]Ellis[/name_u]
[name_u]Evan[/name_u]
[name_u]Fiadh[/name_u]
[name_u]Finlay[/name_u]
[name_u]Garnet[/name_u]
[name_u]Haven[/name_u]
[name_u]Hudson[/name_u]
[name_u]Indie[/name_u]/Indy
[name_u]James[/name_u]
[name_u]Jasper[/name_u]
[name_u]Jude[/name_u]
[name_u]Kit[/name_u]
[name_u]Lennon[/name_u]
[name_u]Lane[/name_u]
[name_u]Luca[/name_u]
[name_u]Maverick[/name_u]
[name_u]Piper[/name_u]
[name_u]Quinn[/name_u]
[name_u]Rey[/name_u]
[name_u]Reid[/name_u]
[name_u]Sage[/name_u]
[name_u]Shea[/name_u]/Shay
[name_u]Sidney[/name_u]/Sydney
[name_u]Spencer[/name_u]
[name_u]Theo[/name_u]
[name_u]Wren[/name_u]
What about…
Sailor
[name_u]River[/name_u]
[name_u]Harbor[/name_u]/Arbor
[name_u]Sky[/name_u]
[name_u]Hadley[/name_u]
[name_u]Blake[/name_u]
[name_u]Blair[/name_u]
[name_u]Hunter[/name_u]
[name_u]Parker[/name_u]
[name_u]Valentine[/name_u]
[name_u]Quinn[/name_u]
[name_u]Madison[/name_u]
[name_u]Mackenzie[/name_u]
River is a nice suggestion, but too similar to the last name. 