Functional Abdominal Pain: This condition is characterized by recurrent abdominal pain that is not explained by other medical conditions. The pain is often related to school or other stressors, and the fact that the episodes occur while going to and coming back from school, but not during holidays, suggests a strong psychological component. The mother's history of migraine also supports this diagnosis, as there is often a familial component to functional gastrointestinal disorders.

Motion Sickness: Given that the episodes occur while the child is on the school bus, motion sickness is a plausible explanation. The symptoms of motion sickness, including nausea, vomiting, and abdominal discomfort, align with the child's presentation.
School Phobia or Anxiety: The child's symptoms may be a manifestation of anxiety or fear related to attending school. This could be due to various factors such as bullying, academic pressures, or separation anxiety.
Irritable Bowel Syndrome (IBS): Although less likely given the specific pattern of symptoms related to school travel, IBS could be considered, especially if the child experiences changes in bowel habits or if the abdominal pain is associated with defecation.

Inflammatory Bowel Disease (IBD): Conditions like Crohn's disease or ulcerative colitis can present with abdominal pain and vomiting. Although the pattern of symptoms does not strongly suggest IBD, it is crucial to rule out these conditions due to their potential for serious complications.
Gastroesophageal Reflux Disease (GERD): GERD can cause abdominal pain and vomiting, especially if the symptoms worsen after eating or at night. However, the specific timing related to school bus travel makes this less likely.
Intestinal Obstruction or Other Surgical Conditions: Although rare, conditions like intussusception or malrotation can cause intermittent abdominal pain and vomiting. These are medical emergencies and must be considered, even if the likelihood is low.

Cyclic Vomiting Syndrome: A rare condition characterized by recurrent episodes of severe vomiting that are separated by symptom-free periods. The episodes can be triggered by stress, among other factors.
Abdominal Migraine: A condition that presents with recurrent episodes of abdominal pain, often accompanied by other migraine symptoms like headache or nausea. Given the mother's history of migraine, this could be considered, although it is less common.