As the parent of a teen who is near getting his off, I am curious why you think this person needs a care package for this. Will this person's parents not provide what he/she needs? If not... that's wonderful that you want to do this. If so, it could be overkill.
In the event you are concerned they will not have what they need, here are what have been helpful for my son:
1) Soft foods for a few days after the first few visits: Yogurt, soft pasta, jello, shakes, smoothies. (Fairly difficult to do this unless you are in the home with the person).
2) Advil: (please don't dose someone else's kid with medicine, you may not be aware of a drug interaction, and if it were MY kid, I'd be ticked that medications were made available to my kid without my knowledge).
3) Dental Wax: This is supplied by the orthodontist and offered at every visit, so no need to buy their own really. Unless they just like having spare packs all around everywhere. My kid has only used it a handful of times, total, in 20 months of braces.
4) Wisps (the little disposable toothbrushes): He likes these for school, or other times he isn't at home to brush after eating. It keeps his brackets free of visually obvious debris-- a big deal to hormone laden teens.
5) A little sympathy. THIS you can supply in hugs, smiles, and a good ear to listen without saying "it isn't that bad...."
Also, avoid things---like eating things in front of him/her that they aren't supposed to eat: sticky candy, chewing gum, popcorn, nuts, corn on the cobb, meat on the bone,
If you really want to go big: get her a waterpik for cleaning her teeth. :)