There are a few things going on here.
First, there is the no alcohol/substance rule that the parent/guardian (and child) agrees to when participating on the sports team. Sometimes these rules are imposed by the school or district itself, but more often they are imposed by a independent non-profit that manages state level competitions. So if your V-ball team wants to make it to the State Championships, all your team members have to abide by these rules. The school often winds up enforcing these rules, but not setting them. I don’t know the case law about such enforcement, but since the rules have been around for decades, my guess is that its been found legal. (I don’t know if this would apply to 8th grade sports, so it may not be applicable in this situation.)
There is also the compelling interest of the state itself to prevent alcohol or drug abuse. If the student needed to be hospitalized after this incident, that is clearly alcohol abuse (as opposed to a parent giving her a glass of wine under supervision). The state can argue that the nature of the incident, appearing intoxicating before a group of students (I presume from the mother claiming it involved other students on snapchat) could result in peer pressure among other students to abuse alcohol, and therefore the state has a compelling interest to regulate this particular off-campus activity.
Second, there is a potentially illegal act – in many jurisdictions underage drinking is illegal unless the alcohol was provided by a parent/guardian, under their direct supervision. This might present a sword of Damocles to the mother: if she provided the alcohol and admits to this fact, she could be charged with reckless endangerment and lose custody.(1) In precious few situations is illegal activity protected speech. Flag burning is one. Violating public nuisance ordinances at a protest may be another. Rob a bank to protest late stage capitalism, you’ll be arrested for bank robbery, not your views on US economic policy.
Finally, the mother is claiming that the school violated her daughter’s free speech rights. The problem with this argument is that there has to be some connection between the activity and the speech being alleged suppressed. As was noted by others here, and the court, drinking alcohol by itself is not protected speech. Even if she were protesting drinking age laws by taking a drink, it would be a stretch to say this was protected, since the same protest could be made without taking a drink, and certainly without drinking until she blacked out. (2)
As others have said here: I hope the girl gets help with her drinking problem.
(1) This was not implied in the article, and I am not trying to imply the mother provided the alcohol, just trying to point out that it may be legal to provide alcohol to one’s child/charge, but there are limits to that legality.
(2) The article states the mother claimed that other students encouraged her daughter to drink until blacking out. If this is the case, then an investigation into who did the encouraging should be conducted. N.C.'s life was put at risk, and the perpetrators should face the consequences of their reckless endangerment.