I highly recommend a horse that is about 7-10 years old for a beginning rider. By the age of 10 the horse is usually over it's juvenille antics, is usually calm and has alot of experience under it's belt (or saddle
Just about any breed is good but I would stay away from more hyper horses like Arabians. With that being said, not all Arabians are "hot", but in my 26 years of riding and being around horses I found Arabians can be a little more hyper and usually not a good choice for beginners. Quarter Horses, Appaloosas, and Paints are usually good choices but in the end it all depends on the horse itself. You could come across a wonderful, laid back, "bomb proof" Arabian that has been used for trail riding all of it's life and also see a wound up, barrel racing Quarter Horse. It all depends on the animal, how they were trained, and what they have been used for.
Also, don't fall into buying a young horse for a young, beginning rider with the thought of "They can grow up and learn together". I've seen that so many times and usually the child gets injured and/or loses interest because they just don't have the ability or knowledge to train a young horse.
Stay away from auctions because you normally find lame or sick horses that are so doped up to hide illness and you get them home and are stuck with a nag with a ton of problems.
Look at alot of horses. Ride the horse yourself and then have your daughter ride it. If you have a riding instructor or a friend with a ton of horse knowledge, take them with you and get their opinion.
Look at how the horse moves. Is it a smooth ride? Generally horses with longer necks ride smoother than a horse with a short neck. Horses with shorter backs normally have less back problems than a horse with a long back.
Look closely at the hooves. Any deep cracks? Are they flat footed? Have they foundered? A bad hoof can mean pain, less time being able to ride and larger farrier bills for corrective shoeing. No hoof, no horse.
After riding the horse stick around and observe them in the pasture or stall. Do they chew? Are the fence posts and trees chewed up? Do they crib? Do they weave back and forth in the stall out of boredom? Do they chew on the stall walls, or worse, on theirself? If they have pasture mates, are they aggressive toward them? Offer them some grain if they're around other horses. Do they pin back their ears and get aggressive with you or the other horses?
That's it for now but I'm sure I'll think of other things in a little while.