Which of the following sentences is grammatically correct?

The sentence "Everyone should bring his or her own lunch" is grammatically correct because it uses the singular pronoun "his or her" to refer back to the singular indefinite pronoun "everyone." In English, "everyone" is treated as a singular noun, which necessitates a singular pronoun to maintain agreement between the subject and the pronoun.

Using "his or her" is a traditional way to create a gender-neutral reference that includes all individuals, acknowledging that "everyone" may refer to both males and females. This structure is widely accepted as grammatically correct in formal writing and is appropriate in this context.

Other options deviate from this standard: the use of "their" with a singular antecedent can be considered informal and somewhat controversial, depending on the context. "One's" is also a singular reference but may sound a bit stilted to some readers. Lastly, "they" is a plural pronoun, so "they own lunch" does not correctly match the singular subject "everyone." Therefore, the option that utilizes "his or her" accurately aligns both subject and pronoun in number.