Jack B. Sowards
Nicholas Meyer was an unofficial screenwriter, and we talk about him in the "Director" section. But officially, the screenwriter was Jack B. Sowards, a television writer best known for his work in Westerns like Bonanza. Star Trek II was his only movie script, but since it was based on a television show, it made perfect sense. (Source)
He also managed to bring one of the film's key players back to make it. Sowards was brought in in part to talk to Leonard Nimoy, who had had his fill of the pointy ears and was eager put Star Trek behind him. The producers worked and worked, but Nimoy kept saying no. This Enterprise mission was going to go forward without its first officer, which could have been disastrous.
But then came Sowards, who had a good relationship with Nimoy, and called him up to promise the one thing he hadn't had before: a brilliant death scene for Mr. Spock. Not only would Nimoy get a terrific moment, but he'd also never have to play Mr. Spock again.
Well Nimoy bought it, and the rest is history. Sowards wrote the scene, Spock's death left all of us weeping like children, and Star Trek got the high point of the whole phenomenon. It was so powerful, in fact, that it caused Nimoy to rethink his attitude toward Spock. He returned to direct Star Trek III and would reprise his role for numerous additional appearances. We have Sowards to thank for that.
He made other great contributions to the script too, such as the Kobayashi Maru concept, but nothing quite tops getting the Vulcan back. Sowards continued working in TV for another eight years before retiring in 1990. He passed away in 2007, having put in motion one of the most beloved death scenes ever put on film.