Is a girl's fasting valid if she intended to fast on the first night of Ramadan but forgot to specify the intention to fast before Fajr due to her preoccupation with her grandfather's death, knowing that she had suhoor with the intention of fasting?

Intention (niyyah) is the resolve and determination of the heart to perform an action. So, whenever the heart resolves to fast at any part of the night, then one has made the intention. The pre-dawn meal (suhoor) for the purpose of fasting is considered an intention. Therefore, if you intended to fast and partook in the suhoor, this is sufficient, and no make-up fasts are required of you.