She was advised to monitor her health due to a history of diabetes in her family.

a 是不定冠词，表示"一个"或"某种"。 history 是名词，意思是"历史、记录"，这里指家族成员的健康状况记录。 of 是介词，表示所属关系，连接"糖尿病"和"家族历史"。 a history of diabetes in her family 表示"她的家族有糖尿病病史"，即家族中曾有成员患过糖尿病。