Is the son sinful who delayed his mother's Fajr prayer, because he thought he would return early to book a medical appointment, but the time was delayed until after noon?

The time for the Fajr prayer is from the appearance of true dawn until sunrise. It is not permissible to delay the prayer beyond this time except for a valid excuse, such as sleep or forgetfulness. In such cases, it must be made up immediately once the excuse is no longer present. As for missing the prayer due to fear of missing a doctor's appointment, this is not a valid excuse. This is considered a grave sin that requires repentance and a firm resolve not to repeat it.