Seconding @nath_kai, I have yet to play a game that tops 2064: Read Only Memories in terms of presentation and choice of pronouns. Just letting you write your own pronouns in is a pretty simple way to include everyone. Barring that option, I think the best solution is a trinary of he, she, and they. It’s the most commonly recognized non-binary pronoun, and the simplest to implement.
Just be sure to do more than a quick copy+paste. Fallout 4 had that problem with her/hers.
Supergiant’s upcoming game, Pyre, allows the player to choose between she/her, he/him, and they/them (first couple of minutes of this video) in the same style the game offers other choices about the player’s background in a way I rather liked.
I’d also say one thing that’s worth noting: not everyone who uses he/him or she/her pronouns IDs as a man or a woman. There are a lot of nonbinary people who, for one reason or another, prefer ‘binary’ pronouns. But many games tie the pronoun and gender together inseparably, which can be alienating. Where possible, I’d stick to genderless titles and epithets in general gameplay. Ser/Serah in place of Sir/Madam, Regent in place of King or Queen, and so on.