There has to be a pronoun-antecedent agreement for the sentences.
The correct sentence is D. Some of the children forgot their notebooks
"Some children" is plural and so is "their."
It's not A because "everyone" is a singular indefinite pronoun and "their" is plural.
For B, "neither" is singular and "their" is plural.
And for C, "none" is singular and "were" is plural.
