The mother needs to make a formal request in writing if she wants the school to perform an evaluation, and the school has a special form for this called Permission to Evaluate Form. She can also write a letter with very specific wording. Either way, the form or the letter need to have the name of the evaluation that she wants done and why.
It would help if her pediatrician agrees that the child requires an evaluation and refers her to a specialist for a neurological-psychological evaluation. If she suspects ADHD then she should request a Pediatric Neurologist and NOT just have the pediatrician start medicating her daughter or agree with her own personal assessment. The diagnosis (or more than one diagnosis) will carry more weight that way. She HAS TO HAVE the evaluation done the right way. If it's ADHD and she's right, great... then she and the doctor and the school can all work together. But if it's not then she'll have to face that it's her parenting.
Also know that if the child has ADHD, the child is a girl and it does present differently in girls than in boys. There may also be some Oppositional Defiance, which is a common comorbid diagnosis as well as Sensory Integration Disorder.
I have a daughter with severe ADHD, ODD, and SID. I was the opposite of your relative... I did whatever it took to avoid that diagnosis until I couldn't ignore it any more. I put all blame on myself and my daughter.
But if her mother has a, uh, lifestyle then it sounds like she's not a stable force in her daughter's life. It sounds from the insinuations that it really is a parenting issue more than a child issue. The child's behavior that you describe sounds like frustration and behavior she likely has witnessed her mother behaving. She would need consistent discipline and a mother that doesn't blame others for her own faults and failings in life and in parenting... who doesn't take constructive criticism as a personal attack.