Some great brands for entry-level guitars are: Yamaha, Luna, Fender, Jasmine. Higher end guitars (brands like Martin and Taylor) have entry models, too, and you'll pay more for these but also reap the benefits: wonderful sounding guitars that feel great in the hands. (The "Little Martin" and the "Baby Taylor" leave shelves at roughly $600. Great deals, given that their sound is bigger than their size and price, and you'll want to keep them forever).
That being said, you can go cheap and still be fine. When we attend conventions, the guitar at our booth is a Yamaha that we bought for $170. It sounds and feels great! Besides, most new guitarists won't know the difference between a $200 and a $2,000 guitar. (Here's a $2,000 tip: put custom-light or extra-light gauge coated strings on that $150 guitar and your child may think it's better than the costly Martin or Taylor. It's not, of course, but they won't know this until they've been playing a few years.) That being said, we still don't recommend buying something less than $125 or $150 or so. You’ll start to get what you pay for: the guitar won’t sound great and it will constantly go out of tune.
Another option is to rent a guitar at a local music store. In Lexington, KY (home for us), guitar rentals are pretty cheap: around $14/month or so. This is a great option while you're still trying to figure out what kind of guitar to buy. (Be sure to check out This Article for guidance on guitar sizing.)
So bottom line, a beginner can't appreciate a super-expensive instrument for what it is. But it's equally unwise to saddle a new student with a poorly built instrument. As a teacher, I've witnessed the latter scenario many times: a student is passionate about guitar, improving daily, writing their own songs, and they just have a real knack for all of it...but their $100 guitar sounds like it rejects the whole project--constantly going out of tune and sounding thin and lifeless.
But a good sounding, well-built guitar is inspiring to play, and doesn't have to break the bank. Present it to the ambitious student they'll thank you forever 😊