Option 1 -> "on behalf of" means representing someone, not working for their benefit.
Option 2 -> "by way of" means "via" or "as a form of," which doesn't fit the context.
Option 3 -> "in case of" implies a conditional event, which is illogical here.
Option 4 -> "for the sake of" means for the benefit or welfare of someone, fitting perfectly.
Hence, Option 4: for the sake of -> This phrase correctly conveys that Anokhi worked hard for the benefit and welfare of her children's future, ensuring they had opportunities. -> correct