In the sentence ‘Everyone should make everyone’s part’, what adjective possessive pronoun should replace the word everyone’s?
"Everyone" is singular, so the following are all acceptable:
"Her or his" or "her" would sometimes be used to combat male-positive gender bias. "Her" or "his" might also be used in a non-offensive way when referring to a group made up only of females or males. The last option, "their", is still shunned by some purists, but is gaining sway as a gender-neutral singular possessive pronoun.