What is the ruling regarding a brother who was suckled by a mother several times, but she had no milk at the time, and she was only suckling him to quiet him?

If the aforementioned person is not a brother by suckling, and he suckled from your mother without there being any milk in her breast, then this suckling has no legal validity. This child remains a stranger to the household, and their daughters must observe hijab in his presence, unless there is another reason for mahramiyyah.