If you are on a budget and are looking for a good beginner flute, you have a couple of options: a used metal flute that’s a good brand or a new/used Nuvo plastic composite flute.
For metal flutes, I do recommend getting a used student model flute with a C Foot, closed keys, and a good brand name. For smaller arms, I highly recommend a curved headjoint.
You can get some work done on a used instrument and it will ultimately be better than a new non-reputable brand. Buying a brand that isn’t well-made will cost more in repairs in the long run and will be impossible to produce a decent sound.
Some music stores typically have some used instruments or might be able to point you in the right direction. Perhaps you know someone who played the flute in high school who no longer plays or has upgraded their flute. Sometimes you can get lucky and find a used flute that’s a good brand online and then fix it up. Brands to look for include Yamaha, Geminhardt, Armstrong, or Bundy.
Another great option if you want something new is the Nuvo Jflute. It’s a plastic composite that gets a nice sound, is lightweight, is waterproof, and doesn’t break easily. The J-headjoint is shorter and is much easier for shorter arms. The Nuvo flute is amazing for children just starting out or someone playing just for fun, but I would recommend getting a standard metal one if they decided to play in school band or wanted to become more involved with the flute.
Good luck and I hope that helps! Let me know if you decide on anything or if you have any questions.
*This information is based on my personal experience and I am in no way being compensated for these recommendations.