Yeppers. Every horse will be "THE ONE" (cue dramatic choir music). In the long run, though, compatability is important enough that it will be worth it to have your daughter along to test each prospect (after the seller/owner tries the horse in front of you, after your experienced horseperson test the horse, in that order. If the owner/seller won't get on the horse, RED FLAG). Plus, she'll always remember the experience of horse hunting.
Here is a list of good questions to ask (with appropriate answers for a first time owner). The basic, whole list is from http://www.ultimatehorsesite.com/info/questions.html
, but I did tweak the answers a bit, some were a tad vague.
1. Mare or Gelding?: This was probably in the sale ad, but either gender is fine for a beginner owner. Mares can be goofy when they come in season, so if you look at a mare, you might ask how "marish" she tends to get.
2. Is he registered, and, if so, do you have the original papers?: Registry status wouldn't matter to me, as the saying goes, you don't ride the papers, but if your daughter is interested in a specific breed for showing, or any other reason, see the original papers at the time you see the horse. All should be up to date with registry paperwork.
3. What is the horse's age?: Over ten is a good benchmark for a first horse. Much younger than that and they can still be a bit immature at times. For a kid's first horse, the upper cutoff I would call about 23 years old. An older horse is likely to have more miles under it and be calmer, but there are health concerns and other factors that go into the maintenance of a senior horse.
4. What Condition is he in?: Basically, fit. At least, not terribly over or underweight. You don't need to start horse ownership with a project.
5. Does he have any special needs or health problems?: No. Your vet check should catch most of these. Your vet should not advise you on "buy/don't buy." The vet is there to tell you what is and what isn't. It's your job to decide if the horse's flaws can be lived with or to move on to the next prospect.
6. What is his personality/temperament like?: Everyone has their own definition of spirited, and, honestly, none of those definitions are quite what you need in a first horse. Look for keywords in ads like calm, willing, quiet, bombproof, kid-safe, 4-H horse, youth horse, etc.
7. How tall is he?: Not a big concern, but should "match" your daughter. If you or any other family members plan on riding, take that into account.
8. What kind of training does the horse have?: That should match what your daughter plans on doing with the horse. Ex. Don't buy a trained barrel horse if she plans to ride hunters, don't buy a western pleasure horse if she wants to run barrels. For a general fun saddle horse, a solid foundation and tons of experience in general riding is fine, no discipline specific training neccessary.
9. Is it trail/road safe?: Should be a resounding yes. The first horse should have a lot of experience in a broad range of circumstances.
10. Who trained it?: Doesn't really matter, unless you want to call a trainer and get more background on the horse.
The horse should ALWAYS: Stand tied, load easily into a trailer, stand for the farrier and vet, stand for bathing/grooming, nicely lift his feet for cleaning/maintenance, be easy to catch, etc. Your daughter will likely be doing many of these daily things herself, and it will be easier for all involved if the horse is easy to live with on the ground.
Good luck, and happy horsehunting!