Not only my brothers but also mom loves to drench their omelets in ketchup.
which pronoun should replace their in the sentence above?
question 2 options:
his
her
his and her
no change is necessary. my choice

You are correct - no change is indeed necessary in this sentence, because the pronoun their is the correct pronoun which you should use in the example above.
You cannot use his, because there are more than one brother involved. You cannot use her, because the omelette belongs to both the mother and the brothers. And the third option is also incorrect because of his in the singular form. 

The first choice

Explanation:

jacqueline never expected to see her name on a book therefore you can tell she never would've thought she'd be successful