I am in massive agreement with Saskia! Wonderful post!
Buying horses is all about knowing your limits and priorities and sticking to them like glue. Sit down with your daughter and her instructor, and make a list of traits that you absolutely must have, don't mind one way or the other, and cannot have in your daughter's horse. For example, you might say something like "The horse must be solid at all three gaits and stand for the farrier, a bay would be nice but that doesn't really matter, and we really do not want a horse that cannot stand tied." Take your list with you when you go look at horses, and objectively compare the prospect to the ideal. Between the three of you you should be able to come up with a good and realistic list. When you know exactly what you are looking for, do not compromise or settle. "THE" horse is out there!!
I also highly recommend a vet check before purchasing, particularly if you're going to be shopping in the higher price brackets. It isn't fun to spend several thousand dollars on a horse, and six months down the road discover that he's not sound enough to be ridden, and needs many more thousands in vet bills and maintenance. The fee that your local equine vet will charge for an exam is a pittance compared to the costs associated with chronic health/soundness issues.
Along with that, it's wise to look at a horse at least twice before buying. Make an appointment the first time, but the second time just stop by. Oftentimes that will head off any opportunity for a dishonest seller to misrepresent the horse.
One more thing to consider is leasing a horse for a few months. That will let you and your daughter ease into the realities and responsibilities of horse ownership without all of the risk associated with a full purchase. If a lease goes well, then take the next step up to a purchase.
Good luck, and happy horse-shopping!